title: Dirac equation

text: The equation also implied the existence of a new form of matter, "antimatter", previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a "theoretical" justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin. The wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation.


title: Dirac equation

text: Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics. This accomplishment has been described as fully on a par with the works of Newton, Maxwell, and Einstein before him. In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin- particles.


title: Dirac equation

text: The Dirac equation appears on the floor of Westminster Abbey on the plaque commemorating Paul Dirac's life, which was unveiled on 13 November 1995.


title: Dirac equation

text: In its modern formulation for field theory, the Dirac equation is written in terms of a Dirac spinor field formula_1 taking values in a complex vector space described concretely as formula_2, defined on flat spacetime (Minkowski space) formula_3. Its expression also contains gamma matrices and a parameter formula_4 interpreted as the mass, as well as other physical constants.


title: Dirac equation

text: The gamma matrices are a set of four formula_6 complex matrices (elements of formula_7) which satisfy the defining "anti"-commutation relations:


title: Dirac equation

text: These matrices can be realized explicitly under a choice of representation. Two common choices are the Dirac representation


title: Dirac equation

text: where formula_10 are the Pauli matrices, and the chiral representation: the formula_11 are the same, but


title: Dirac equation

text: where formula_14 is a four-vector (often it is the four-vector differential operator formula_15). The summation over the index formula_16 is implied.


title: Dirac equation

text: Using the property of gamma matrices (which follows straightforwardly from Hermicity properties of the formula_19) that


title: Dirac equation

text: one can derive the adjoint Dirac equation by taking the Hermitian conjugate of the Dirac equation and multiplying on the right by formula_21:


title: Dirac equation

text: where the partial derivative acts from the right on formula_23: written in the usual way in terms of a left action of the derivative, we have


title: Dirac equation

text: Another approach to derive this expression is by variational methods, applying Noether's theorem for the global formula_28 symmetry to derive the conserved current formula_29


title: Dirac equation

text: This immediately simplifies as there are no partial derivatives of formula_30 in the Lagrangian. formula_31 is the infinitesimal variation


title: Dirac equation

text: Since the Dirac operator acts on 4-tuples of square-integrable functions, its solutions should be members of the same Hilbert space. The fact that the energies of the solutions do not have a lower bound is unexpected.


title: Dirac equation

text: After picking a representation for the gamma matrices formula_19, solving this is a matter of solving a system of linear equations. It is a representation-free property of gamma matrices that the solution space is two-dimensional (see here).


title: Dirac equation

text: For example, in the chiral representation for formula_19, the solution space is parametrised by a formula_42 vector formula_43, with


title: Dirac equation

text: Both the Dirac equation and the Adjoint Dirac equation can be obtained from (varying) the action with a specific Lagrangian density that is given by:


title: Dirac equation

text: If one varies this with respect to formula_1 one gets the adjoint Dirac equation. Meanwhile, if one varies this with respect to formula_30 one gets the Dirac equation.


title: Dirac equation

text: For this action, the conserved current formula_50 above arises as the conserved current corresponding to the global formula_28 symmetry through Noether's theorem for field theory. Gauging this field theory by changing the symmetry to a local, spacetime point dependent one gives gauge symmetry (really, gauge redundancy). The resultant theory is quantum electrodynamics or QED. See below for a more detailed discussion.


title: Dirac equation

text: The Dirac equation is invariant under Lorentz transformations, that is, under the action of the Lorentz group formula_52 or strictly formula_53, the component connected to the identity.


title: Dirac equation

text: For a Dirac spinor viewed concretely as taking values in formula_2, the transformation under a Lorentz transformation formula_55 is given by a formula_56 complex matrix formula_57. There are some subtleties in defining the corresponding formula_57, as well as a standard abuse of notation.


title: Dirac equation

text: Most treatments occur at the Lie algebra level. For a more detailed treatment see here. The Lorentz group of formula_6 "real" matrices acting on formula_3 is generated by a set of six matrices formula_61 with components


title: Dirac equation

text: When both the formula_63 indices are raised or lowered, these are simply the 'standard basis' of antisymmetric matrices.


title: Dirac equation

text: This is an abuse of notation, but a standard one. The reason is formula_57 is not a well-defined function of formula_55, since there are two different sets of components formula_68 (up to equivalence) which give the same formula_55 but different formula_57. In practice we implicitly pick one of these formula_68 and then formula_57 is well defined in terms of formula_78


title: Dirac equation

text: Associated to Lorentz invariance is a conserved Noether current, or rather a tensor of conserved Noether currents formula_81. Similarly, since the equation is invariant under translations, there is a tensor of conserved Noether currents formula_82, which can be identified as the stress-energy tensor of the theory. The Lorentz current formula_81 can be written in terms of the stress-energy tensor in addition to a tensor representing internal angular momentum.


title: Dirac equation

text: The Dirac equation was also used (historically) to define a quantum-mechanical theory where formula_17 is instead interpreted as a wave-function.


title: Dirac equation

text: where is the wave function for the electron of rest mass with spacetime coordinates . The are the components of the momentum, understood to be the momentum operator in the Schrödinger equation. Also, is the speed of light, and is the reduced Planck constant. These fundamental physical constants reflect special relativity and quantum mechanics, respectively.


title: Dirac equation

text: Dirac's purpose in casting this equation was to explain the behavior of the relativistically moving electron, and so to allow the atom to be treated in a manner consistent with relativity. His rather modest hope was that the corrections introduced this way might have a bearing on the problem of atomic spectra.


title: Dirac equation

text: Up until that time, attempts to make the old quantum theory of the atom compatible with the theory of relativity, which were based on discretizing the angular momentum stored in the electron's possibly non-circular orbit of the atomic nucleus, had failed – and the new quantum mechanics of Heisenberg, Pauli, Jordan, Schrödinger, and Dirac himself had not developed sufficiently to treat this problem. Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of matter and introduced new mathematical classes of objects that are now essential elements of fundamental physics.


title: Dirac equation

text: The new elements in this equation are the four matrices , , and , and the four-component wave function . There are four components in because the evaluation of it at any given point in configuration space is a bispinor. It is interpreted as a superposition of a spin-up electron, a spin-down electron, a spin-up positron, and a spin-down positron.


title: Dirac equation

text: These matrices and the form of the wave function have a deep mathematical significance. The algebraic structure represented by the gamma matrices had been created some 50 years earlier by the English mathematician W. K. Clifford. In turn, Clifford's ideas had emerged from the mid-19th-century work of the German mathematician Hermann Grassmann in his "Lineare Ausdehnungslehre" ("Theory of Linear Extensions"). The latter had been regarded as almost incomprehensible by most of his contemporaries. The appearance of something so seemingly abstract, at such a late date, and in such a direct physical manner, is one of the most remarkable chapters in the history of physics.


title: Dirac equation

text: The single symbolic equation thus unravels into four coupled linear first-order partial differential equations for the four quantities that make up the wave function. The equation can be written more explicitly in Planck units as:


title: Dirac equation

text: which makes it clearer that it is a set of four partial differential equations with four unknown functions.


title: Dirac equation

text: The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically as they do in the Maxwell equations that govern the behavior of light — the equations must be differentially of the "same order" in space and time. In relativity, the momentum and the energies are the space and time parts of a spacetime vector, the four-momentum, and they are related by the relativistically invariant relation


title: Dirac equation

text: which says that the length of this four-vector is proportional to the rest mass . Substituting the operator equivalents of the energy and momentum from the Schrödinger theory produces the Klein–Gordon equation describing the propagation of waves, constructed from relativistically invariant objects,


title: Dirac equation

text: with the wave function being a relativistic scalar: a complex number which has the same numerical value in all frames of reference. Space and time derivatives both enter to second order. This has a telling consequence for the interpretation of the equation. Because the equation is second order in the time derivative, one must specify initial values both of the wave function itself and of its first time-derivative in order to solve definite problems. Since both may be specified more or less arbitrarily, the wave function cannot maintain its former role of determining the probability density of finding the electron in a given state of motion. In the Schrödinger theory, the probability density is given by the positive definite expression


title: Dirac equation

text: The fact that the density is positive definite and convected according to this continuity equation implies that one may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law. A proper relativistic theory with a probability density current must also share this feature. To maintain the notion of a convected density, one must generalize the Schrödinger expression of the density and current so that space and time derivatives again enter symmetrically in relation to the scalar wave function. The Schrödinger expression can be kept for the current, but the probability density must be replaced by the symmetrically formed expression


title: Dirac equation

text: which now becomes the 4th component of a spacetime vector, and the entire probability 4-current density has the relativistically covariant expression


title: Dirac equation

text: The continuity equation is as before. Everything is compatible with relativity now, but the expression for the density is no longer positive definite; the initial values of both and may be freely chosen, and the density may thus become negative, something that is impossible for a legitimate probability density. Thus, one cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.


title: Dirac equation

text: Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of quantum field theory, where it is known as the Klein–Gordon equation, and describes a spinless particle field (e.g. pi meson or Higgs boson). Historically, Schrödinger himself arrived at this equation before the one that bears his name but soon discarded it. In the context of quantum field theory, the indefinite density is understood to correspond to the "charge" density, which can be positive or negative, and not the probability density.


title: Dirac equation

text: Dirac thus thought to try an equation that was "first order" in both space and time. One could, for example, formally (i.e. by abuse of notation) take the relativistic expression for the energy


title: Dirac equation

text: replace by its operator equivalent, expand the square root in an infinite series of derivative operators, set up an eigenvalue problem, then solve the equation formally by iterations. Most physicists had little faith in such a process, even if it were technically possible.


title: Dirac equation

text: As the story goes, Dirac was staring into the fireplace at Cambridge, pondering this problem, when he hit upon the idea of taking the square root of the wave operator thus:


title: Dirac equation

text: On multiplying out the right side it is apparent that, in order to get all the cross-terms such as to vanish, one must assume


title: Dirac equation

text: Dirac, who had just then been intensely involved with working out the foundations of Heisenberg's matrix mechanics, immediately understood that these conditions could be met if , , and are "matrices", with the implication that the wave function has "multiple components". This immediately explained the appearance of two-component wave functions in Pauli's phenomenological theory of spin, something that up until then had been regarded as mysterious, even to Pauli himself. However, one needs at least matrices to set up a system with the properties required — so the wave function had "four" components, not two, as in the Pauli theory, or one, as in the bare Schrödinger theory. The four-component wave function represents a new class of mathematical object in physical theories that makes its first appearance here.


title: Dirac equation

text: Taking formula_104 shows that all the components of the wave function "individually" satisfy the relativistic energy–momentum relation. Thus the sought-for equation that is first-order in both space and time is


title: Dirac equation

text: To demonstrate the relativistic invariance of the equation, it is advantageous to cast it into a form in which the space and time derivatives appear on an equal footing. New matrices are introduced as follows:


title: Dirac equation

text: and the equation takes the form (remembering the definition of the covariant components of the 4-gradient and especially that )


title: Dirac equation

text: where there is an implied summation over the values of the twice-repeated index , and is the 4-gradient. In practice one often writes the gamma matrices in terms of 2 × 2 sub-matrices taken from the Pauli matrices and the 2 × 2 identity matrix. Explicitly the standard representation is


title: Dirac equation

text: denotes the anticommutator. These are the defining relations of a Clifford algebra over a pseudo-orthogonal 4-dimensional space with metric signature . The specific Clifford algebra employed in the Dirac equation is known today as the Dirac algebra. Although not recognized as such by Dirac at the time the equation was formulated, in hindsight the introduction of this "geometric algebra" represents an enormous stride forward in the development of quantum theory.


title: Dirac equation

text: The Dirac equation may now be interpreted as an eigenvalue equation, where the rest mass is proportional to an eigenvalue of the 4-momentum operator, the proportionality constant being the speed of light:


title: Dirac equation

text: Using formula_113 (formula_114 is pronounced "d-slash"), according to Feynman slash notation, the Dirac equation becomes:


title: Dirac equation

text: In practice, physicists often use units of measure such that , known as natural units. The equation then takes the simple form


title: Dirac equation

text: A fundamental theorem states that if two distinct sets of matrices are given that both satisfy the Clifford relations, then they are connected to each other by a similarity transformation:


title: Dirac equation

text: The transformation is unique up to a multiplicative factor of absolute value 1. Let us now imagine a Lorentz transformation to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector. For the operator to remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index. These new gammas will themselves satisfy the Clifford relations, because of the orthogonality of the Lorentz transformation. By the fundamental theorem, one may replace the new set by the old set subject to a unitary transformation. In the new frame, remembering that the rest mass is a relativistic scalar, the Dirac equation will then take the form


title: Dirac equation

text: then the transformed Dirac equation is produced in a way that demonstrates manifest relativistic invariance:


title: Dirac equation

text: Thus, settling on any unitary representation of the gammas is final, provided the spinor is transformed according to the unitary transformation that corresponds to the given Lorentz transformation.


title: Dirac equation

text: The various representations of the Dirac matrices employed will bring into focus particular aspects of the physical content in the Dirac wave function. The representation shown here is known as the "standard" representation – in it, the wave function's upper two components go over into Pauli's 2 spinor wave function in the limit of low energies and small velocities in comparison to light.


title: Dirac equation

text: The considerations above reveal the origin of the gammas in "geometry", hearkening back to Grassmann's original motivation; they represent a fixed basis of unit vectors in spacetime. Similarly, products of the gammas such as represent "oriented surface elements", and so on. With this in mind, one can find the form of the unit volume element on spacetime in terms of the gammas as follows. By definition, it is


title: Dirac equation

text: For this to be an invariant, the epsilon symbol must be a tensor, and so must contain a factor of , where is the determinant of the metric tensor. Since this is negative, that factor is "imaginary". Thus


title: Dirac equation

text: This matrix is given the special symbol , owing to its importance when one is considering improper transformations of space-time, that is, those that change the orientation of the basis vectors. In the standard representation, it is


title: Dirac equation

text: It takes a leading role when questions of "parity" arise because the volume element as a directed magnitude changes sign under a space-time reflection. Taking the positive square root above thus amounts to choosing a handedness convention on spacetime.


title: Dirac equation

text: The necessity of introducing half-integer spin goes back experimentally to the results of the Stern–Gerlach experiment. A beam of atoms is run through a strong inhomogeneous magnetic field, which then splits into parts depending on the intrinsic angular momentum of the atoms. It was found that for silver atoms, the beam was split in two; the ground state therefore could not be integer, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into three parts, corresponding to atoms with . The conclusion is that silver atoms have net intrinsic angular momentum of . Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so in SI units: (Note that bold faced characters imply Euclidean vectors in 3 dimensions, whereas the Minkowski four-vector can be defined as formula_125.)


title: Dirac equation

text: Here and formula_127 represent the components of the electromagnetic four-potential in their standard SI units, and the three sigmas are the Pauli matrices. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual classical Hamiltonian of a charged particle interacting with an applied field in SI units:


title: Dirac equation

text: This Hamiltonian is now a matrix, so the Schrödinger equation based on it must use a two-component wave function. On introducing the external electromagnetic 4-vector potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form:


title: Dirac equation

text: A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by , have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the gyromagnetic ratio of the electron, standing in front of Pauli's new term, is explained from first principles. This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness. There is more however. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. First the equation is written in the form of coupled equations for 2-spinors with the SI units restored:


title: Dirac equation

text: Assuming the field is weak and the motion of the electron non-relativistic, the total energy of the electron is approximately equal to its rest energy, and the momentum going over to the classical value,


title: Dirac equation

text: which is of order – thus at typical energies and velocities, the bottom components of the Dirac spinor in the standard representation are much suppressed in comparison to the top components. Substituting this expression into the first equation gives after some rearrangement


title: Dirac equation

text: The operator on the left represents the particle energy reduced by its rest energy, which is just the classical energy, so one can recover Pauli's theory upon identifying his 2-spinor with the top components of the Dirac spinor in the non-relativistic approximation. A further approximation gives the Schrödinger equation as the limit of the Pauli theory. Thus, the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities. This also was a great triumph for the new equation, as it traced the mysterious that appears in it, and the necessity of a complex wave function, back to the geometry of spacetime through the Dirac algebra. It also highlights why the Schrödinger equation, although superficially in the form of a diffusion equation, actually represents the propagation of waves.


title: Dirac equation

text: It should be strongly emphasized that this separation of the Dirac spinor into large and small components depends explicitly on a low-energy approximation. The entire Dirac spinor represents an "irreducible" whole, and the components just neglected here to arrive at the Pauli theory will bring in new phenomena in the relativistic regime – antimatter and the idea of creation and annihilation of particles.


title: Dirac equation

text: In the massless case formula_135, the Dirac equation reduces to the Weyl equation, which describes relativistic massless spin- particles.


title: Dirac equation

text: The critical physical question in a quantum theory is this: what are the physically observable quantities defined by the theory? According to the postulates of quantum mechanics, such quantities are defined by Hermitian operators that act on the Hilbert space of possible states of a system. The eigenvalues of these operators are then the possible results of measuring the corresponding physical quantity. In the Schrödinger theory, the simplest such object is the overall Hamiltonian, which represents the total energy of the system. To maintain this interpretation on passing to the Dirac theory, the Hamiltonian must be taken to be


title: Dirac equation

text: where, as always, there is an implied summation over the twice-repeated index . This looks promising, because one can see by inspection the rest energy of the particle and, in the case of , the energy of a charge placed in an electric potential . What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is


title: Dirac equation

text: Thus, the Dirac Hamiltonian is fundamentally distinguished from its classical counterpart, and one must take great care to correctly identify what is observable in this theory. Much of the apparently paradoxical behavior implied by the Dirac equation amounts to a misidentification of these observables.


title: Dirac equation

text: The negative solutions to the equation are problematic, for it was assumed that the particle has a positive energy. Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions. Since they exist, they cannot simply be ignored, for once the interaction between the electron and the electromagnetic field is included, any electron placed in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy. Real electrons obviously do not behave in this way, or they would disappear by emitting energy in the form of photons.


title: Dirac equation

text: To cope with this problem, Dirac introduced the hypothesis, known as hole theory, that the vacuum is the many-body quantum state in which all the negative-energy electron eigenstates are occupied. This description of the vacuum as a "sea" of electrons is called the Dirac sea. Since the Pauli exclusion principle forbids electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy eigenstate, and positive-energy electrons would be forbidden from decaying into negative-energy eigenstates.


title: Dirac equation

text: If an electron is forbidden from simultaneously occupying positive-energy and negative-energy eigenstates, then the feature known as Zitterbewegung, which arises from the interference of positive-energy and negative-energy states, would have to be considered to be an unphysical prediction of time-dependent Dirac theory. This conclusion may be inferred from the explanation of hole theory given in the preceding paragraph. Recent results have been published in Nature [R. Gerritsma, G. Kirchmair, F. Zaehringer, E. Solano, R. Blatt, and C. Roos, Nature 463, 68-71 (2010)] in which the Zitterbewegung feature was simulated in a trapped-ion experiment. This experiment impacts the hole interpretation if one infers that the physics-laboratory experiment is not merely a check on the mathematical correctness of a Dirac equation solution but the measurement of a real effect whose detectability in electron physics is still beyond reach.


title: Dirac equation

text: Dirac further reasoned that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate – called a hole – would behave like a positively charged particle. The hole possesses a "positive" energy because energy is required to create a particle–hole pair from the vacuum. As noted above, Dirac initially thought that the hole might be the proton, but Hermann Weyl pointed out that the hole should behave as if it had the same mass as an electron, whereas the proton is over 1800 times heavier. The hole was eventually identified as the positron, experimentally discovered by Carl Anderson in 1932.


title: Dirac equation

text: It is not entirely satisfactory to describe the "vacuum" using an infinite sea of negative-energy electrons. The infinitely negative contributions from the sea of negative-energy electrons have to be canceled by an infinite positive "bare" energy and the contribution to the charge density and current coming from the sea of negative-energy electrons is exactly canceled by an infinite positive "jellium" background so that the net electric charge density of the vacuum is zero. In quantum field theory, a Bogoliubov transformation on the creation and annihilation operators (turning an occupied negative-energy electron state into an unoccupied positive energy positron state and an unoccupied negative-energy electron state into an occupied positive energy positron state) allows us to bypass the Dirac sea formalism even though, formally, it is equivalent to it.


title: Dirac equation

text: In certain applications of condensed matter physics, however, the underlying concepts of "hole theory" are valid. The sea of conduction electrons in an electrical conductor, called a Fermi sea, contains electrons with energies up to the chemical potential of the system. An unfilled state in the Fermi sea behaves like a positively charged electron, though it is referred to as a "hole" rather than a "positron". The negative charge of the Fermi sea is balanced by the positively charged ionic lattice of the material.


title: Dirac equation

text: In quantum field theories such as quantum electrodynamics, the Dirac field is subject to a process of second quantization, which resolves some of the paradoxical features of the equation.


title: Dirac equation

text: The Dirac equation is Lorentz covariant. Articulating this helps illuminate not only the Dirac equation, but also the Majorana spinor and Elko spinor, which although closely related, have subtle and important differences.


title: Dirac equation

text: Understanding Lorentz covariance is simplified by keeping in mind the geometric character of the process. Let formula_139 be a single, fixed point in the spacetime manifold. Its location can be expressed in multiple coordinate systems. In the physics literature, these are written as formula_140 and formula_141, with the understanding that both formula_140 and formula_141 describe "the same" point formula_139, but in different local frames of reference (a frame of reference over a small extended patch of spacetime).


title: Dirac equation

text: One can imagine formula_139 as having a fiber of different coordinate frames above it. In geometric terms, one says that spacetime can be characterized as a fiber bundle, and specifically, the frame bundle. The difference between two points formula_140 and formula_141 in the same fiber is a combination of rotations and Lorentz boosts. A choice of coordinate frame is a (local) section through that bundle.


title: Dirac equation

text: Coupled to the frame bundle is a second bundle, the spinor bundle. A section through the spinor bundle is just the particle field (the Dirac spinor, in the present case). Different points in the spinor fiber correspond to the same physical object (the fermion) but expressed in different Lorentz frames. Clearly, the frame bundle and the spinor bundle must be tied together in a consistent fashion to get consistent results; formally, one says that the spinor bundle is the associated bundle; it is associated to a principal bundle, which in the present case is the frame bundle. Differences between points on the fiber correspond to the symmetries of the system. The spinor bundle has two distinct generators of its symmetries: the total angular momentum and the intrinsic angular momentum. Both correspond to Lorentz transformations, but in different ways.


title: Dirac equation

text: The presentation here follows that of Itzykson and Zuber. It is very nearly identical to that of Bjorken and Drell. A similar derivation in a general relativistic setting can be found in Weinberg. Here we fix our spacetime to be flat, that is, our spacetime is Minkowski space.


title: Dirac equation

text: where formula_152 parameterizes the Lorentz transformation, and formula_153 are the six 4×4 matrices satisfying:


title: Dirac equation

text: This matrix can be interpreted as the intrinsic angular momentum of the Dirac field. That it deserves this interpretation arises by contrasting it to the generator formula_155 of Lorentz transformations, having the form


title: Dirac equation

text: Note the formula_140 above does "not" have a prime on it: the above is obtained by transforming formula_159 obtaining the change to formula_160 and then returning to the original coordinate system formula_161.


title: Dirac equation

text: The geometrical interpretation of the above is that the frame field is affine, having no preferred origin. The generator formula_155 generates the symmetries of this space: it provides a relabelling of a fixed point formula_163 The generator formula_153 generates a movement from one point in the fiber to another: a movement from formula_159 with both formula_140 and formula_141 still corresponding to the same spacetime point formula_168 These perhaps obtuse remarks can be elucidated with explicit algebra.


title: Dirac equation

text: If the Dirac equation is to be covariant, then it should have exactly the same form in all Lorentz frames:


title: Dirac equation

text: The two spinors formula_1 and formula_173 should both describe the same physical field, and so should be related by a transformation that does not change any physical observables (charge, current, mass, "etc.") The transformation should encode only the change of coordinate frame. It can be shown that such a transformation is a 4×4 unitary matrix. Thus, one may presume that the relation between the two frames can be written as


title: Dirac equation

text: An explicit expression for formula_178 (equal to the expression given above) can be obtained by considering a Lorentz transformation of infinitesimal rotation near the indentity transfomation:


title: Dirac equation

text: where formula_180 is the metric tensor : formula_181 and is symmteric while formula_182 is antisymmetric. After plugging and chugging, one obtains


title: Dirac equation

text: which is the (infinitesimal) form for formula_150 above and yields the relation formula_185 . To obtain the affine relabelling, write


title: Dirac equation

text: After properly antisymmetrizing, one obtains the generator of symmetries formula_155 given earlier. Thus, both formula_155 and formula_153 can be said to be the "generators of Lorentz transformations", but with a subtle distinction: the first corresponds to a relabelling of points on the affine frame bundle, which forces a translation along the fiber of the spinor on the spin bundle, while the second corresponds to translations along the fiber of the spin bundle (taken as a movement formula_159 along the frame bundle, as well as a movement formula_191 along the fiber of the spin bundle.) Weinberg provides additional arguments for the physical interpretation of these as total and intrinsic angular momentum.


title: Dirac equation

text: This article has developed the Dirac equation in flat spacetime according to special relativity. It is possible to formulate the Dirac equation in curved spacetime.


title: Dirac equation

text: This article developed the Dirac equation using four vectors and Schrödinger operators. The Dirac equation in the algebra of physical space uses a Clifford algebra over the real numbers, a type of geometric algebra.


title: Dirac equation

text: Natural units are used in this section. The coupling constant is labelled by convention with formula_192: this parameter can also be viewed as modelling the electron charge.


title: Dirac equation

text: This is a global symmetry, known as the formula_28 vector symmetry (as opposed to the formula_28 axial symmetry: see below). By Noether's theorem there is a corresponding conserved current: this has been mentioned previously as


title: Dirac equation

text: If we 'promote' the global symmetry, parametrised by the constant formula_199, to a local symmetry, parametrised by a function formula_200, or equivalently formula_201 the Dirac equation is no longer invariant: there is a residual derivative of formula_202.


title: Dirac equation

text: The fix proceeds as in scalar electrodynamics: the partial derivative is promoted to a covariant derivative formula_203


title: Dirac equation

text: The covariant derivative depends on the field being acted on. The newly introduced formula_206 is the 4-vector potential from electrodynamics, but also can be viewed as a formula_28 gauge field, or a formula_28 connection.


title: Dirac equation

text: but can also be derived by asking that covariant derivatives transform under a gauge transformation as


title: Dirac equation

text: We then obtain a gauge-invariant Dirac action by promoting the partial derivative to a covariant one:


title: Dirac equation

text: The final step needed to write down a gauge-invariant Lagrangian is to add a Maxwell Lagrangian term,


title: Dirac equation

text: Massless Dirac fermions, that is, fields formula_17 satisfying the Dirac equation with formula_135, admit a second, inequivalent formula_28 symmetry.


title: Dirac equation

text: This is seen most easily by writing the four-component Dirac fermion formula_17 as a pair of two-component vector fields,


title: Dirac equation

text: The earlier vector symmetry is still present, where formula_228 and formula_229 rotate identically. This form of the action makes the second inequivalent formula_28 symmetry manifest:


title: Dirac equation

text: This isn't the only formula_28 symmetry possible, but it is conventional. Any 'linear combination' of the vector and axial symmetries is also a formula_28 symmetry.


title: Dirac equation

text: Classically, the axial symmetry admits a well-formulated gauge theory. But at the quantum level, there is an anomaly, that is, an obstruction to gauging.


title: Dirac equation

text: We can extend this discussion from an abelian formula_28 symmetry to a general non-abelian symmetry under a gauge group formula_237, the group of color symmetries for a theory.


title: Dirac equation

text: Before this section, formula_17 could be viewed as a spinor field on Minkowski space, in other words a function formula_241, and its components in formula_2 are labelled by spin indices, conventionally Greek indices taken from the start of the alphabet formula_243.


title: Dirac equation

text: Promoting the theory to a gauge theory, informally formula_1 acquires a part transforming like formula_239, and these are labelled by color indices, conventionally Latin indices formula_246. In total, formula_17 has formula_248 components, given in indices by formula_249. The 'spinor' labels only how the field transforms under spacetime transformations.


title: Dirac equation

text: Gauging proceeds similarly to the abelian formula_28 case, with a few differences. Under a gauge transformation formula_253 the spinor fields transform as


title: Dirac equation

text: Writing down a gauge-invariant action proceeds exactly as with the formula_28 case, replacing the Maxwell Lagrangian with the Yang–Mills Lagrangian


title: Dirac equation

text: For physical applications, the case formula_267 describes the quark sector of the Standard model which models strong interactions. Quarks are modelled as Dirac spinors; the gauge field is the gluon field. The case formula_268 describes part of the electroweak sector of the Standard model. Leptons such as electrons and neutrinos are the Dirac spinors; the gauge field is the formula_269 gauge boson.


title: Dirac equation

text: This expression can be generalised to arbitrary Lie group formula_237 with connection formula_206 and a representation formula_272, where the colour part of formula_1 is valued in formula_274. Formally, the Dirac field is a function formula_275


title: Dirac equation

text: where here we view formula_280 as a Lie algebra representation of the Lie algebra formula_281 associated to formula_237.


title: Dirac equation

text: This theory can be generalised to curved spacetime, but there are subtleties which arise in gauge theory on a general spacetime (or more generally still, a manifold) which, on flat spacetime, can be ignored. This is ultimately due to the contractibility of flat spacetime which allows us to view a gauge field and gauge transformations as defined globally on formula_3.


title: Cosmic ray

text: Cosmic rays are high-energy particles or clusters of particles (primarily represented by protons or atomic nuclei) that move through space at nearly the speed of light. They originate from the Sun, from outside of the Solar System in our own galaxy, and from distant galaxies. Upon impact with Earth's atmosphere, cosmic rays produce showers of secondary particles, some of which reach the surface, although the bulk is deflected off into space by the magnetosphere or the heliosphere.


title: Cosmic ray

text: Cosmic rays were discovered by Victor Hess in 1912 in balloon experiments, for which he was awarded the 1936 Nobel Prize in Physics.


title: Cosmic ray

text: Direct measurement of cosmic rays, especially at lower energies, has been possible since the launch of the first satellites in the late 1950s. Particle detectors similar to those used in nuclear and high-energy physics are used on satellites and space probes for research into cosmic rays.


title: Cosmic ray

text: Data from the Fermi Space Telescope (2013) have been interpreted as evidence that a significant fraction of primary cosmic rays originate from the supernova explosions of stars. Based on observations of neutrinos and gamma rays from blazar TXS 0506+056 in 2018, active galactic nuclei also appear to produce cosmic rays.


title: Cosmic ray

text: The term "ray" is somewhat of a misnomer, as cosmic rays were, originally, incorrectly believed to be mostly electromagnetic radiation. In common scientific usage, high-energy particles with intrinsic mass are known as "cosmic" rays, while photons, which are quanta of electromagnetic radiation (and so have no intrinsic mass) are known by their common names, such as "gamma rays" or "X-rays", depending on their photon energy.


title: Cosmic ray

text: Of primary cosmic rays, which originate outside of Earth's atmosphere, about 99% are the bare nuclei of well-known atoms (stripped of their electron shells), and about 1% are solitary electrons (that is, one type of beta particle). Of the nuclei, about 90% are simple protons (i.e., hydrogen nuclei); 9% are alpha particles, identical to helium nuclei; and 1% are the nuclei of heavier elements, called HZE ions. These fractions vary highly over the energy range of cosmic rays. A very small fraction are stable particles of antimatter, such as positrons or antiprotons. The precise nature of this remaining fraction is an area of active research. An active search from Earth orbit for anti-alpha particles has failed to detect them.


title: Cosmic ray

text: Upon striking the atmosphere, cosmic rays violently burst atoms into other bits of matter, producing large amounts of pions and muons (which have a short half-life) as well as neutrinos. The neutron composition of the particle cascade increases at lower elevations, reaching between 40% to 80% of the radiation at aircraft altitudes.


title: Cosmic ray

text: Cosmic rays attract great interest practically, due to the damage they inflict on microelectronics and life outside the protection of an atmosphere and magnetic field, and scientifically, because the energies of the most energetic ultra-high-energy cosmic rays have been observed to approach [that's a little more than 21 million times the design energy of particles accelerated by the Large Hadron Collider = (CERN, 2021)]. One can show that such enormous energies might be achieved by means of the centrifugal mechanism of acceleration in active galactic nuclei. At , the highest-energy ultra-high-energy cosmic rays (such as the OMG particle recorded in 1991) have energies comparable to the kinetic energy of a baseball. As a result of these discoveries, there has been interest in investigating cosmic rays of even greater energies. Most cosmic rays, however, do not have such extreme energies; the energy distribution of cosmic rays peaks at .


title: Cosmic ray

text: After the discovery of radioactivity by Henri Becquerel in 1896, it was generally believed that atmospheric electricity, ionization of the air, was caused only by radiation from radioactive elements in the ground or the radioactive gases or isotopes of radon they produce. Measurements of increasing ionization rates at increasing heights above the ground during the decade from 1900 to 1910 could be explained as due to absorption of the ionizing radiation by the intervening air.


title: Cosmic ray

text: In 1909, Theodor Wulf developed an electrometer, a device to measure the rate of ion production inside a hermetically sealed container, and used it to show higher levels of radiation at the top of the Eiffel Tower than at its base. However, his paper published in "Physikalische Zeitschrift" was not widely accepted. In 1911, Domenico Pacini observed simultaneous variations of the rate of ionization over a lake, over the sea, and at a depth of 3 metres from the surface. Pacini concluded from the decrease of radioactivity underwater that a certain part of the ionization must be due to sources other than the radioactivity of the Earth.


title: Cosmic ray

text: In 1912, Victor Hess carried three enhanced-accuracy Wulf electrometers to an altitude of 5,300 metres in a free balloon flight. He found the ionization rate increased approximately fourfold over the rate at ground level. Hess ruled out the Sun as the radiation's source by making a balloon ascent during a near-total eclipse. With the moon blocking much of the Sun's visible radiation, Hess still measured rising radiation at rising altitudes. He concluded that "The results of the observations seem most likely to be explained by the assumption that radiation of very high penetrating power enters from above into our atmosphere." In 1913–1914, Werner Kolhörster confirmed Victor Hess's earlier results by measuring the increased ionization enthalpy rate at an altitude of 9 km. Hess received the Nobel Prize in Physics in 1936 for his discovery.


title: Cosmic ray

text: In the late 1920s and early 1930s the technique of self-recording electroscopes carried by balloons into the highest layers of the atmosphere or sunk to great depths under water was brought to an unprecedented degree of perfection by the German physicist Erich Regener and his group. To these scientists we owe some of the most accurate measurements ever made of cosmic-ray ionization as a function of altitude and depth.


title: Cosmic ray

text: Ernest Rutherford stated in 1931 that "thanks to the fine experiments of Professor Millikan and the even more far-reaching experiments of Professor Regener, we have now got for the first time, a curve of absorption of these radiations in water which we may safely rely upon".


title: Cosmic ray

text: In the 1920s, the term "cosmic rays" was coined by Robert Millikan who made measurements of ionization due to cosmic rays from deep under water to high altitudes and around the globe. Millikan believed that his measurements proved that the primary cosmic rays were gamma rays; i.e., energetic photons. And he proposed a theory that they were produced in interstellar space as by-products of the fusion of hydrogen atoms into the heavier elements, and that secondary electrons were produced in the atmosphere by Compton scattering of gamma rays. But then, sailing from Java to the Netherlands in 1927, Jacob Clay found evidence, later confirmed in many experiments, that cosmic ray intensity increases from the tropics to mid-latitudes, which indicated that the primary cosmic rays are deflected by the geomagnetic field and must therefore be charged particles, not photons. In 1929, Bothe and Kolhörster discovered charged cosmic-ray particles that could penetrate 4.1 cm of gold. Charged particles of such high energy could not possibly be produced by photons from Millikan's proposed interstellar fusion process.


title: Cosmic ray

text: In 1930, Bruno Rossi predicted a difference between the intensities of cosmic rays arriving from the east and the west that depends upon the charge of the primary particles—the so-called "east-west effect". Three independent experiments found that the intensity is, in fact, greater from the west, proving that most primaries are positive. During the years from 1930 to 1945, a wide variety of investigations confirmed that the primary cosmic rays are mostly protons, and the secondary radiation produced in the atmosphere is primarily electrons, photons and muons. In 1948, observations with nuclear emulsions carried by balloons to near the top of the atmosphere showed that approximately 10% of the primaries are helium nuclei (alpha particles) and 1% are nuclei of heavier elements such as carbon, iron, and lead.


title: Cosmic ray

text: During a test of his equipment for measuring the east-west effect, Rossi observed that the rate of near-simultaneous discharges of two widely separated Geiger counters was larger than the expected accidental rate. In his report on the experiment, Rossi wrote "... it seems that once in a while the recording equipment is struck by very extensive showers of particles, which causes coincidences between the counters, even placed at large distances from one another." In 1937 Pierre Auger, unaware of Rossi's earlier report, detected the same phenomenon and investigated it in some detail. He concluded that high-energy primary cosmic-ray particles interact with air nuclei high in the atmosphere, initiating a cascade of secondary interactions that ultimately yield a shower of electrons, and photons that reach ground level.


title: Cosmic ray

text: Soviet physicist Sergei Vernov was the first to use radiosondes to perform cosmic ray readings with an instrument carried to high altitude by a balloon. On 1 April 1935, he took measurements at heights up to 13.6 kilometres using a pair of Geiger counters in an anti-coincidence circuit to avoid counting secondary ray showers.


title: Cosmic ray

text: Homi J. Bhabha derived an expression for the probability of scattering positrons by electrons, a process now known as Bhabha scattering. His classic paper, jointly with Walter Heitler, published in 1937 described how primary cosmic rays from space interact with the upper atmosphere to produce particles observed at the ground level. Bhabha and Heitler explained the cosmic ray shower formation by the cascade production of gamma rays and positive and negative electron pairs.


title: Cosmic ray

text: Measurements of the energy and arrival directions of the ultra-high-energy primary cosmic rays by the techniques of "density sampling" and "fast timing" of extensive air showers were first carried out in 1954 by members of the Rossi Cosmic Ray Group at the Massachusetts Institute of Technology. The experiment employed eleven scintillation detectors arranged within a circle 460 metres in diameter on the grounds of the Agassiz Station of the Harvard College Observatory. From that work, and from many other experiments carried out all over the world, the energy spectrum of the primary cosmic rays is now known to extend beyond 10 eV. A huge air shower experiment called the Auger Project is currently operated at a site on the Pampas of Argentina by an international consortium of physicists. The project was first led by James Cronin, winner of the 1980 Nobel Prize in Physics from the University of Chicago, and Alan Watson of the University of Leeds, and later by scientists of the international Pierre Auger Collaboration. Their aim is to explore the properties and arrival directions of the very highest-energy primary cosmic rays. The results are expected to have important implications for particle physics and cosmology, due to a theoretical Greisen–Zatsepin–Kuzmin limit to the energies of cosmic rays from long distances (about 160 million light years) which occurs above 10 eV because of interactions with the remnant photons from the Big Bang origin of the universe. Currently the Pierre Auger Observatory is undergoing an upgrade to improve its accuracy and find evidence for the yet unconfirmed origin of the most energetic cosmic rays.


title: Cosmic ray

text: High-energy gamma rays (>50MeV photons) were finally discovered in the primary cosmic radiation by an MIT experiment carried on the OSO-3 satellite in 1967. Components of both galactic and extra-galactic origins were separately identified at intensities much less than 1% of the primary charged particles. Since then, numerous satellite gamma-ray observatories have mapped the gamma-ray sky. The most recent is the Fermi Observatory, which has produced a map showing a narrow band of gamma ray intensity produced in discrete and diffuse sources in our galaxy, and numerous point-like extra-galactic sources distributed over the celestial sphere.


title: Cosmic ray

text: Early speculation on the sources of cosmic rays included a 1934 proposal by Baade and Zwicky suggesting cosmic rays originated from supernovae. A 1948 proposal by Horace W. Babcock suggested that magnetic variable stars could be a source of cosmic rays. Subsequently, Sekido "et al." (1951) identified the Crab Nebula as a source of cosmic rays. Since then, a wide variety of potential sources for cosmic rays began to surface, including supernovae, active galactic nuclei, quasars, and gamma-ray bursts.


title: Cosmic ray

text: Later experiments have helped to identify the sources of cosmic rays with greater certainty. In 2009, a paper presented at the International Cosmic Ray Conference by scientists at the Pierre Auger Observatory in Argentina showed ultra-high energy cosmic rays originating from a location in the sky very close to the radio galaxy Centaurus A, although the authors specifically stated that further investigation would be required to confirm Centaurus A as a source of cosmic rays. However, no correlation was found between the incidence of gamma-ray bursts and cosmic rays, causing the authors to set upper limits as low as 3.4 × 10× erg·cm on the flux of cosmic rays from gamma-ray bursts.


title: Cosmic ray

text: In 2009, supernovae were said to have been "pinned down" as a source of cosmic rays, a discovery made by a group using data from the Very Large Telescope. This analysis, however, was disputed in 2011 with data from PAMELA, which revealed that "spectral shapes of [hydrogen and helium nuclei] are different and cannot be described well by a single power law", suggesting a more complex process of cosmic ray formation. In February 2013, though, research analyzing data from "Fermi" revealed through an observation of neutral pion decay that supernovae were indeed a source of cosmic rays, with each explosion producing roughly 3 × 10 – 3 × 10J of cosmic rays.


title: Cosmic ray

text: Supernovae do not produce all cosmic rays, however, and the proportion of cosmic rays that they do produce is a question which cannot be answered without deeper investigation. To explain the actual process in supernovae and active galactic nuclei that accelerates the stripped atoms, physicists use shock front acceleration as a plausibility argument (see picture at right).


title: Cosmic ray

text: In 2017, the Pierre Auger Collaboration published the observation of a weak anisotropy in the arrival directions of the highest energy cosmic rays. Since the Galactic Center is in the deficit region, this anisotropy can be interpreted as evidence for the extragalactic origin of cosmic rays at the highest energies. This implies that there must be a transition energy from galactic to extragalactic sources, and there may be different types of cosmic-ray sources contributing to different energy ranges.


title: Cosmic ray

text: Cosmic rays originate as primary cosmic rays, which are those originally produced in various astrophysical processes. Primary cosmic rays are composed mainly of protons and alpha particles (99%), with a small amount of heavier nuclei (≈1%) and an extremely minute proportion of positrons and antiprotons. Secondary cosmic rays, caused by a decay of primary cosmic rays as they impact an atmosphere, include photons, leptons, and hadrons, such as electrons, positrons, muons, and pions. The latter three of these were first detected in cosmic rays.


title: Cosmic ray

text: Primary cosmic rays mostly originate from outside the Solar System and sometimes even outside the Milky Way. When they interact with Earth's atmosphere, they are converted to secondary particles. The mass ratio of helium to hydrogen nuclei, 28%, is similar to the primordial elemental abundance ratio of these elements, 24%. The remaining fraction is made up of the other heavier nuclei that are typical nucleosynthesis end products, primarily lithium, beryllium, and boron. These nuclei appear in cosmic rays in much greater abundance (≈1%) than in the solar atmosphere, where they are only about 10 as abundant as helium. Cosmic rays composed of charged nuclei heavier than helium are called HZE ions. Due to the high charge and heavy nature of HZE ions, their contribution to an astronaut's radiation dose in space is significant even though they are relatively scarce.


title: Cosmic ray

text: This abundance difference is a result of the way in which secondary cosmic rays are formed. Carbon and oxygen nuclei collide with interstellar matter to form lithium, beryllium and boron in a process termed cosmic ray spallation. Spallation is also responsible for the abundances of scandium, titanium, vanadium, and manganese ions in cosmic rays produced by collisions of iron and nickel nuclei with interstellar matter.


title: Cosmic ray

text: At high energies the composition changes and heavier nuclei have larger abundances in some energy ranges. Current experiments aim at more accurate measurements of the composition at high energies.


title: Cosmic ray

text: Satellite experiments have found evidence of positrons and a few antiprotons in primary cosmic rays, amounting to less than 1% of the particles in primary cosmic rays. These do not appear to be the products of large amounts of antimatter from the Big Bang, or indeed complex antimatter in the universe. Rather, they appear to consist of only these two elementary particles, newly made in energetic processes.


title: Cosmic ray

text: Preliminary results from the presently operating Alpha Magnetic Spectrometer ("AMS-02") on board the International Space Station show that positrons in the cosmic rays arrive with no directionality. In September 2014, new results with almost twice as much data were presented in a talk at CERN and published in Physical Review Letters. A new measurement of positron fraction up to 500 GeV was reported, showing that positron fraction peaks at a maximum of about 16% of total electron+positron events, around an energy of . At higher energies, up to 500 GeV, the ratio of positrons to electrons begins to fall again. The absolute flux of positrons also begins to fall before 500 GeV, but peaks at energies far higher than electron energies, which peak about 10 GeV. These results on interpretation have been suggested to be due to positron production in annihilation events of massive dark matter particles.


title: Cosmic ray

text: Cosmic ray antiprotons also have a much higher average energy than their normal-matter counterparts (protons). They arrive at Earth with a characteristic energy maximum of 2 GeV, indicating their production in a fundamentally different process from cosmic ray protons, which on average have only one-sixth of the energy.


title: Cosmic ray

text: There is no evidence of complex antimatter atomic nuclei, such as antihelium nuclei (i.e., anti-alpha particles), in cosmic rays. These are actively being searched for. A prototype of the "AMS-02" designated "AMS-01", was flown into space aboard the on STS-91 in June 1998. By not detecting any antihelium at all, the "AMS-01" established an upper limit of for the antihelium to helium flux ratio.


title: Cosmic ray

text: When cosmic rays enter the Earth's atmosphere, they collide with atoms and molecules, mainly oxygen and nitrogen. The interaction produces a cascade of lighter particles, a so-called air shower secondary radiation that rains down, including x-rays, protons, alpha particles, pions, muons, electrons, neutrinos, and neutrons. All of the secondary particles produced by the collision continue onward on paths within about one degree of the primary particle's original path.


title: Cosmic ray

text: Typical particles produced in such collisions are neutrons and charged mesons such as positive or negative pions and kaons. Some of these subsequently decay into muons and neutrinos, which are able to reach the surface of the Earth. Some high-energy muons even penetrate for some distance into shallow mines, and most neutrinos traverse the Earth without further interaction. Others decay into photons, subsequently producing electromagnetic cascades. Hence, next to photons, electrons and positrons usually dominate in air showers. These particles as well as muons can be easily detected by many types of particle detectors, such as cloud chambers, bubble chambers, water-Cherenkov or scintillation detectors. The observation of a secondary shower of particles in multiple detectors at the same time is an indication that all of the particles came from that event.


title: Cosmic ray

text: Cosmic rays impacting other planetary bodies in the Solar System are detected indirectly by observing high-energy gamma ray emissions by gamma-ray telescope. These are distinguished from radioactive decay processes by their higher energies above about 10 MeV.


title: Cosmic ray

text: The flux of incoming cosmic rays at the upper atmosphere is dependent on the solar wind, the Earth's magnetic field, and the energy of the cosmic rays. At distances of ≈94 AU from the Sun, the solar wind undergoes a transition, called the termination shock, from supersonic to subsonic speeds. The region between the termination shock and the heliopause acts as a barrier to cosmic rays, decreasing the flux at lower energies (≤ 1 GeV) by about 90%. However, the strength of the solar wind is not constant, and hence it has been observed that cosmic ray flux is correlated with solar activity.


title: Cosmic ray

text: In addition, the Earth's magnetic field acts to deflect cosmic rays from its surface, giving rise to the observation that the flux is apparently dependent on latitude, longitude, and azimuth angle.


title: Cosmic ray

text: The combined effects of all of the factors mentioned contribute to the flux of cosmic rays at Earth's surface. The following table of participial frequencies reach the planet and are inferred from lower-energy radiation reaching the ground.


title: Cosmic ray

text: In the past, it was believed that the cosmic ray flux remained fairly constant over time. However, recent research suggests one-and-a-half- to two-fold millennium-timescale changes in the cosmic ray flux in the past forty thousand years.


title: Cosmic ray

text: The magnitude of the energy of cosmic ray flux in interstellar space is very comparable to that of other deep space energies: cosmic ray energy density averages about one electron-volt per cubic centimetre of interstellar space, or ≈1 eV/cm, which is comparable to the energy density of visible starlight at 0.3 eV/cm, the galactic magnetic field energy density (assumed 3 microgauss) which is ≈0.25 eV/cm, or the cosmic microwave background (CMB) radiation energy density at ≈0.25 eV/cm.


title: Cosmic ray

text: There are two main classes of detection methods. First, the direct detection of the primary cosmic rays in space or at high altitude by balloon-borne instruments. Second, the indirect detection of secondary particle, i.e., extensive air showers at higher energies. While there have been proposals and prototypes for space and balloon-borne detection of air showers, currently operating experiments for high-energy cosmic rays are ground based. Generally direct detection is more accurate than indirect detection. However the flux of cosmic rays decreases with energy, which hampers direct detection for the energy range above 1 PeV. Both direct and indirect detection are realized by several techniques.


title: Cosmic ray

text: Direct detection is possible by all kinds of particle detectors at the ISS, on satellites, or high-altitude balloons. However, there are constraints in weight and size limiting the choices of detectors.


title: Cosmic ray

text: An example for the direct detection technique is a method based on nuclear tracks developed by Robert Fleischer, P. Buford Price, and Robert M. Walker for use in high-altitude balloons. In this method, sheets of clear plastic, like 0.25 mm Lexan polycarbonate, are stacked together and exposed directly to cosmic rays in space or high altitude. The nuclear charge causes chemical bond breaking or ionization in the plastic. At the top of the plastic stack the ionization is less, due to the high cosmic ray speed. As the cosmic ray speed decreases due to deceleration in the stack, the ionization increases along the path. The resulting plastic sheets are "etched" or slowly dissolved in warm caustic sodium hydroxide solution, that removes the surface material at a slow, known rate. The caustic sodium hydroxide dissolves the plastic at a faster rate along the path of the ionized plastic. The net result is a conical etch pit in the plastic. The etch pits are measured under a high-power microscope (typically 1600× oil-immersion), and the etch rate is plotted as a function of the depth in the stacked plastic.


title: Cosmic ray

text: This technique yields a unique curve for each atomic nucleus from 1 to 92, allowing identification of both the charge and energy of the cosmic ray that traverses the plastic stack. The more extensive the ionization along the path, the higher the charge. In addition to its uses for cosmic-ray detection, the technique is also used to detect nuclei created as products of nuclear fission.


title: Cosmic ray

text: There are several ground-based methods of detecting cosmic rays currently in use, which can be divided in two main categories: the detection of secondary particles forming extensive air showers (EAS) by various types of particle detectors, and the detection of electromagnetic radiation emitted by EAS in the atmosphere.


title: Cosmic ray

text: Extensive air shower arrays made of particle detectors measure the charged particles which pass through them. EAS arrays can observe a broad area of the sky and can be active more than 90% of the time. However, they are less able to segregate background effects from cosmic rays than can air Cherenkov telescopes. Most state-of-the-art EAS arrays employ plastic scintillators. Also water (liquid or frozen) is used as a detection medium through which particles pass and produce Cherenkov radiation to make them detectable. Therefore, several arrays use water/ice-Cherenkov detectors as alternative or in addition to scintillators.


title: Cosmic ray

text: By the combination of several detectors, some EAS arrays have the capability to distinguish muons from lighter secondary particles (photons, electrons, positrons). The fraction of muons among the secondary particles is one traditional way to estimate the mass composition of the primary cosmic rays.


title: Cosmic ray

text: An historic method of secondary particle detection still used for demonstration purposes involves the use of cloud chambers to detect the secondary muons created when a pion decays. Cloud chambers in particular can be built from widely available materials and can be constructed even in a high-school laboratory. A fifth method, involving bubble chambers, can be used to detect cosmic ray particles.


title: Cosmic ray

text: More recently, the CMOS devices in pervasive smartphone cameras have been proposed as a practical distributed network to detect air showers from ultra-high-energy cosmic rays. The first app to exploit this proposition was the CRAYFIS (Cosmic RAYs Found in Smartphones) experiment. In 2017, the CREDO (Cosmic Ray Extremely Distributed Observatory) Collaboration released the first version of its completely open source app for Android devices. Since then the collaboration has attracted the interest and support of many scientific institutions, educational institutions and members of the public around the world. Future research has to show in what aspects this new technique can compete with dedicated EAS arrays.


title: Cosmic ray

text: The first detection method in the second category is called the air Cherenkov telescope, designed to detect low-energy (<200 GeV) cosmic rays by means of analyzing their Cherenkov radiation, which for cosmic rays are gamma rays emitted as they travel faster than the speed of light in their medium, the atmosphere. While these telescopes are extremely good at distinguishing between background radiation and that of cosmic-ray origin, they can only function well on clear nights without the Moon shining, have very small fields of view, and are only active for a few percent of the time.


title: Cosmic ray

text: A second method detects the light from nitrogen fluorescence caused by the excitation of nitrogen in the atmosphere by particles moving through the atmosphere. This method is the most accurate for cosmic rays at highest energies, in particular when combined with EAS arrays of particle detectors. Similar to the detection of Cherenkov-light, this method is restricted to clear nights.


title: Cosmic ray

text: Another method detects radio waves emitted by air showers. This technique has a high duty cycle similar to that of particle detectors. The accuracy of this technique was improved in the last years as shown by various prototype experiments, and may become an alternative to the detection of atmospheric Cherenkov-light and fluorescence light, at least at high energies.


title: Cosmic ray

text: Cosmic rays ionize nitrogen and oxygen molecules in the atmosphere, which leads to a number of chemical reactions. Cosmic rays are also responsible for the continuous production of a number of unstable isotopes, such as carbon-14, in the Earth's atmosphere through the reaction:


title: Cosmic ray

text: Cosmic rays kept the level of carbon-14 in the atmosphere roughly constant (70 tons) for at least the past 100,000 years, until the beginning of above-ground nuclear weapons testing in the early 1950s. This fact is used in radiocarbon dating.


title: Cosmic ray

text: Cosmic rays constitute a fraction of the annual radiation exposure of human beings on the Earth, averaging 0.39mSv out of a total of 3mSv per year (13% of total background) for the Earth's population. However, the background radiation from cosmic rays increases with altitude, from 0.3mSv per year for sea-level areas to 1.0mSv per year for higher-altitude cities, raising cosmic radiation exposure to a quarter of total background radiation exposure for populations of said cities. Airline crews flying long distance high-altitude routes can be exposed to 2.2mSv of extra radiation each year due to cosmic rays, nearly doubling their total exposure to ionizing radiation.


title: Cosmic ray

text: Cosmic rays have sufficient energy to alter the states of circuit components in electronic integrated circuits, causing transient errors to occur (such as corrupted data in electronic memory devices or incorrect performance of CPUs) often referred to as "soft errors". This has been a problem in electronics at extremely high-altitude, such as in satellites, but with transistors becoming smaller and smaller, this is becoming an increasing concern in ground-level electronics as well. Studies by IBM in the 1990s suggest that computers typically experience about one cosmic-ray-induced error per 256 megabytes of RAM per month. To alleviate this problem, the Intel Corporation has proposed a cosmic ray detector that could be integrated into future high-density microprocessors, allowing the processor to repeat the last command following a cosmic-ray event. ECC memory is used to protect data against data corruption caused by cosmic rays.


title: Cosmic ray

text: In 2008, data corruption in a flight control system caused an Airbus A330 airliner to twice plunge hundreds of feet, resulting in injuries to multiple passengers and crew members. Cosmic rays were investigated among other possible causes of the data corruption, but were ultimately ruled out as being very unlikely.


title: Cosmic ray

text: In August 2020 scientists reported that ionizing radiation from environmental radioactive materials and cosmic rays may substantially limit the coherence times of qubits if they aren't shielded adequately which may be critical for realizing fault-tolerant superconducting quantum computers in the future.


title: Cosmic ray

text: Galactic cosmic rays are one of the most important barriers standing in the way of plans for interplanetary travel by crewed spacecraft. Cosmic rays also pose a threat to electronics placed aboard outgoing probes. In 2010, a malfunction aboard the "Voyager 2" space probe was credited to a single flipped bit, probably caused by a cosmic ray. Strategies such as physical or magnetic shielding for spacecraft have been considered in order to minimize the damage to electronics and human beings caused by cosmic rays.


title: Cosmic ray

text: On 31 May 2013, NASA scientists reported that a possible crewed mission to Mars may involve a greater radiation risk than previously believed, based on the amount of energetic particle radiation detected by the RAD on the Mars Science Laboratory while traveling from the Earth to Mars in 2011–2012.


title: Cosmic ray

text: Flying high, passengers and crews of jet airliners are exposed to at least 10 times the cosmic ray dose that people at sea level receive. Aircraft flying polar routes near the geomagnetic poles are at particular risk.


title: Cosmic ray

text: Cosmic rays have been implicated in the triggering of electrical breakdown in lightning. It has been proposed that essentially all lightning is triggered through a relativistic process, or "runaway breakdown", seeded by cosmic ray secondaries. Subsequent development of the lightning discharge then occurs through "conventional breakdown" mechanisms.


title: Cosmic ray

text: A role for cosmic rays in climate was suggested by Edward P. Ney in 1959 and by Robert E. Dickinson in 1975. It has been postulated that cosmic rays may have been responsible for major climatic change and mass extinction in the past. According to Adrian Mellott and Mikhail Medvedev, 62-million-year cycles in biological marine populations correlate with the motion of the Earth relative to the galactic plane and increases in exposure to cosmic rays. The researchers suggest that this and gamma ray bombardments deriving from local supernovae could have affected cancer and mutation rates, and might be linked to decisive alterations in the Earth's climate, and to the mass extinctions of the Ordovician.


title: Cosmic ray

text: Danish physicist Henrik Svensmark has controversially argued that because solar variation modulates the cosmic ray flux on Earth, it would consequently affect the rate of cloud formation and hence be an indirect cause of global warming. Svensmark is one of several scientists outspokenly opposed to the mainstream scientific assessment of global warming, leading to concerns that the proposition that cosmic rays are connected to global warming could be ideologically biased rather than scientifically based. Other scientists have vigorously criticized Svensmark for sloppy and inconsistent work: one example is adjustment of cloud data that understates error in lower cloud data, but not in high cloud data; another example is "incorrect handling of the physical data" resulting in graphs that do not show the correlations they claim to show. Despite Svensmark's assertions, galactic cosmic rays have shown no statistically significant influence on changes in cloud cover, and have been demonstrated in studies to have no causal relationship to changes in global temperature.


title: Cosmic ray

text: A handful of studies conclude that a nearby supernova or series of supernovas caused the Pliocene marine megafauna extinction event by substantially increasing radiation levels to hazardous amounts for large seafaring animals.


title: Abhishek Bachchan filmography

text: Abhishek Bachchan is an Indian actor and producer known for his work in Hindi films. He made his debut in 2000, opposite Kareena Kapoor, in the J. P. Dutta-directed war drama "Refugee". His portrayal of the titular unnamed refugee garnered him a nomination for Best Male Debut at the Filmfare Awards ceremony. However, the film was a box office failure, as were a series of his subsequent films, including "Bas Itna Sa Khwaab Hai" (2001), and "Shararat" (2002). His career prospects improved in 2004, when he played a gangster in the Mani Ratnam-directed political drama "Yuva", and a police officer in Sanjay Gadhvi's action thriller "Dhoom". For the former, he won his first Filmfare Award for Best Supporting Actor, and the latter became one of the highest-grossing films of the year and his first commercial success.


title: Abhishek Bachchan filmography

text: In 2005, Bachchan starred in the crime comedy "Bunty Aur Babli" opposite Rani Mukerji, and the Ram Gopal Varma-directed thriller "Sarkar". The former was the second highest-grossing film of the year, and his performance in the latter earned him a second Filmfare Award for Best Supporting Actor. In both films he co-starred with his father, Amitabh Bachchan. That same year, he also appeared in the Bengali film "Antarmahal" (2005). The following year, Bachchan appeared in the Karan Johar-directed musical romantic drama "Kabhi Alvida Naa Kehna" (2006), for which he received his third consecutive Filmfare Award for Best Supporting Actor. The same year, he reunited with Dutta on the historical romance "Umrao Jaan" (2006), and reprised his role as a police officer in the second installment of the "Dhoom" series, entitled "Dhoom 2" (2006). The latter was the highest grossing Bollywood film to that point. Bachchan next starred alongside Aishwarya Rai in "Guru" (2007), a biopic inspired by the life of the businessman Dhirubhai Ambani. His titular role in the film was positively received.


title: Abhishek Bachchan filmography

text: In 2008, Bachchan starred in Varma's "Sarkar Raj", the romantic comedy "Dostana", and the superhero film "Drona". The following year, Bachchan produced and appeared in "Paa" (2009), a comedy-drama which won the National Film Award for Best Feature Film in Hindi. In the same year, he also starred in the Rakeysh Omprakash Mehra-directed satire "Delhi-6" (2009). Bachchan then suffered a brief setback in his career as he starred in another series of films that performed poorly at the box-office, including Ratnam's drama "Raavan" (2010), and the action thriller "Game" (2011). His role in the comedy "Bol Bachchan" (2012), opposite Ajay Devgan, proved to be his first box-office success since "Paa". He followed this with three commercially successful films: "Dhoom 3" (2013), opposite Aamir Khan, the action heist comedy "Happy New Year" (2014), opposite Shah Rukh Khan, and the 2016 film "Housefull 3". Abhishek Bachchan starred in "Manmarziyaan" in 2018, an Indian romantic drama directed by Anurag Kashyap.


title: Abhishek Bachchan filmography

text: He was seen in Red Chillies Entertainment production "Bob Biswas", that went on floors and was released on 3 December 2021.


title: 1984 New York City Subway shooting

text: On December 22, 1984, Bernhard Goetz () shot four young men on a New York City Subway train in Manhattan after they allegedly tried to rob him.


title: 1984 New York City Subway shooting

text: Goetz surrendered to police nine days later and was charged with attempted murder, assault, reckless endangerment, and several firearms offenses. Initially, Goetz was viewed by most as a victim and by some as a vigilante, and he received widespread public recognition and support. A grand jury refused to indict Goetz on the more serious charges, voting indictments only for criminal gun possession. However, public opinion about Goetz wavered due to statements and alleged damaging details of the incident that later were released by the prosecution. Goetz was then re-indicted by a second grand jury on more serious charges. At a later jury trial, he was found guilty of one count of carrying an unlicensed firearm, for which he served eight months of a one-year sentence. In 1996, Darrell Cabey, one of those Goetz shot and who was left paraplegic and brain damaged as a result of his injuries, obtained a civil judgment of $43 million against Goetz.


title: 1984 New York City Subway shooting

text: The incident sparked a nationwide debate on crime in major U.S. cities, the legal limits of self-defense based on Cabey being shot once or twice, and the extent to which the citizenry could rely on the police to secure their safety. Goetz, dubbed the "Subway Vigilante" by the New York press, came to symbolize New Yorkers' frustrations with the high crime rates of the 1980s. Goetz was both praised and vilified in the media and public opinion, and the New York criminal justice system came under increased public scrutiny and criticism. The incident has been cited as a contributing factor to the groundswell movement against urban crime and disorder, and successful National Rifle Association campaigns to loosen restrictions for concealed carrying of firearms.


title: 1984 New York City Subway shooting

text: Bernhard Goetz stated that, three years before the incident, he had been attacked in the Canal Street subway station while transporting electronic equipment by three men who attempted to rob him. The attackers smashed Goetz into a plate-glass door and threw him to the ground, injuring his chest and knee. Goetz assisted an off-duty officer in arresting one of them; the other two attackers escaped. Goetz was angered when the arrested attacker spent less than half the time in the police station than the time spent by Goetz himself, and he was angered further when this attacker was charged only with criminal mischief for ripping Goetz's jacket. Goetz subsequently applied for a permit to carry a concealed handgun, on the basis of routinely carrying valuable equipment and large sums of cash, but his application was denied for insufficient need. He bought a 5-shot .38-caliber revolver during a trip to Florida.


title: 1984 New York City Subway shooting

text: At the time, New York City street crime received little media coverage. N.Y. Governor Mario Cuomo was considered a leading future Democratic presidential candidate. The subway shooting made news around the world, dismal subway train images and city crime received widespread coverage, and the incident had significant political overtones.


title: 1984 New York City Subway shooting

text: —boarded a downtown 2 train (a Broadway–Seventh Avenue express). The teenagers, each of whom had previously been arrested and convicted at least once, stated they were on their way to rob a video arcade in Manhattan. When the train arrived at the 14th Street station in Manhattan at about 1:30PM, fifteen to twenty other passengers remained with them in the R22 subway car, the seventh car of the ten-car train.


title: 1984 New York City Subway shooting

text: Several minutes prior Goetz left his apartment, walking one block to this station on the way downtown to meet some friends for a holiday drink. After the train entered the station Goetz approached the rear door of the 7th car. Canty was lying on the long bench just to the right of this door. Allen was seated to Canty's left, on the short seat on the left side of this door. Ramseur and Cabey were seated across from this door on the short seat by the conductor's cab. A woman passenger was seated several feet from Canty's right.


title: 1984 New York City Subway shooting

text: The door opened, Goetz entered and crossed the aisle, and took a seat on the long bench across from Canty and this woman, approximately centered between them. Simultaneously a male passenger sat down at Goetz's left at the far end of Goetz's bench. The doors closed and the train left for the next express stop at Chambers Street, typically a four-minute ride.


title: 1984 New York City Subway shooting

text: According to Goetz's statement to the Concord police department (New Hampshire), several seconds later Canty asked him, "How are you doing?" Goetz responded, "Fine." Goetz said the group gave signals to each other, and shortly thereafter Canty and Allen rose from their seats and moved over to Goetz's left, blocking him off from the other passengers in the car. Canty then said, "Give me five dollars." Goetz subsequently pulled a handgun and fired 5 shots at the four, wounding all. Canty and Ramseur testified at the criminal trial that they were begging, and had only requested the money, not demanded it. Cabey did not testify and Allen took the Fifth Amendment.


title: 1984 New York City Subway shooting

text: Sources differ in reporting the sequence of shots fired, timing of shots, and whether Cabey was shot once or twice. The following are four versions from significant or reliable sources describing the sequence of shots:


title: 1984 New York City Subway shooting

text: At the Bronx civil trial, Goetz testified the first shot was Canty, Allen second, the third shot missed, Cabey fourth, and Ramseur fifth. The following similar shooting sequence is verbatim from Goetz's website:


title: 1984 New York City Subway shooting

text: Goetz does a dry fire shooting demonstration (five shots in 1.0 seconds at 4 hypothetical targets on both sides of him) on the Biography Channel show "Aftermath with William Shatner".


title: 1984 New York City Subway shooting

text: Multiple criminal trial witnesses testified all shots came in "rapid succession"; Criminal trial medical evidence showed Cabey was shot once in left side, and not shot twice as had been widely reported.


title: 1984 New York City Subway shooting

text: Prior to the criminal trial, the media reported that Cabey had been shot on the fourth shot and then again on the fifth shot, with Goetz saying, "You don't look too bad, here's another," or, "You seem all right, here's another." This sequence of shots was discredited at the criminal trial when it was revealed that one of the five shots had missed and Cabey was shot once in the left side; however, the misconception that Cabey was shot twice occurred long after the criminal trial.


title: 1984 New York City Subway shooting

text: "Speed is everything," Goetz said in a videotaped statement made after he surrendered nine days later. He told police that while still seated, he planned a "pattern of fire" from left to right. He then stood, stepped clear of Canty, drew his revolver, turned back to Canty, and fired four shots, one at each man, then fired a fifth shot. At the civil trial years later he said, "I was trying to get as many of them as I could as quickly as I could." Other sources simply repeated Goetz's statements to New York City police as to the sequence of shots: Canty was shot first, then Allen, then Ramseur, then Cabey. In the related proceeding "People v. Goetz", the New York Court of Appeals summarized the incident:


title: 1984 New York City Subway shooting

text: "It appears from the evidence before the Grand Jury that Canty approached Goetz, possibly with Allen beside him, and stated 'Give me five dollars.' Neither Canty nor any of the other youths displayed a weapon. Goetz responded by standing up, pulling out his handgun, and firing four shots in rapid succession. The first shot hit Canty in the chest; the second struck Allen in the back; the third went through Ramseur's arm and into his left side; the fourth was fired at Cabey, who apparently was then standing in the corner of the car, but missed, deflecting instead off a wall of the conductor's cab. After Goetz briefly surveyed the train scene around him, he fired another shot at Cabey, who then was sitting on the end bench of the car. The bullet entered the rear of Cabey's side and severed his spinal cord."


title: 1984 New York City Subway shooting

text: According to his statements to police, Goetz checked the first two men to make sure that they had been "taken care of," then upon seeing that the fourth man, Cabey, was now sitting down rocking side to side and seemed not seriously hurt, said, "You seem to be all right, here's another," and fired another shot into Cabey's stomach. Cabey actually was shot only once in the left side, a fact not made known to Goetz or his attorneys until shortly before the trial. One bullet had missed, fragmenting on the steel cab wall behind Cabeythis missed shot would also be the basis of a charge of reckless endangerment of other passengers. After the criminal trial Goetz denied shooting Cabey on the 5th shot, saying he had been confused by misinformation.


title: 1984 New York City Subway shooting

text: Goetz said one of the boys made gestures that may have implied he had a weapon. Canty then approached Goetz for the $5. Goetz rose and partly unzipped his jacket where the revolver was concealed, and plotted his "pattern of fire" for shooting them. Goetz asked Canty what he said, who repeated his statement. According to Troy Canty, Goetz then unzipped his jacket the rest of the way, drew the gun, assumed a combat stance gripping the revolver with both hands, and shot Canty through the center of his body. He then turned to shoot Allen who had tried to flee, hitting him in the back, and then shot Ramseur, wounding him in the chest and arm. He then shot again, at Cabey, but may have missed. According to Goetz he then approached Cabey and shot him on the ground; however, another witness disputed that Goetz shot Cabey a second time.


title: 1984 New York City Subway shooting

text: Cabey ended up slumped in the short seat in the corner of the car next to the conductor's cab. Whether Cabey was struck by the fourth shot or by the fifth was critical to Goetz's claim of self-defense and thus an issue fiercely contested at trial. Medical experts testified that such an injury would render the lower half of Cabey's body instantly useless. According to the prosecution, the fourth shot missed; then Goetz shot a seated Cabey at point-blank range with the fifth. The defense theory of how Cabey ended up in the seat was that he was standing when hit by the fourth shot, then collapsed into the seat due to the lurching and swaying of the train, with the fifth shot being the shot that missed.


title: 1984 New York City Subway shooting

text: A summary of Goetz's statements to the police had become public two months after the incident, drawing intense media coverage. Perhaps most damaging to Goetz's public support and to his claim of acting in self-defense was his statement that he had said, "You don't look so bad, here's another," before firing at Cabey a second time.


title: 1984 New York City Subway shooting

text: At trial, one witness testified that Goetz approached to within "two to three feet" of a seated Cabey, then demonstrated how Goetz stood directly in front of Cabey and fired downward shooting Cabey in the stomach, a description that matched Goetz's published statements, but contradicted medical evidence that Cabey was shot once in the left side. Eight other independent witnesses testified that all shots came in "rapid succession"; one of these said the firing lasted "about a second". None of the eight heard a pause before the final shot, and none saw Goetz standing in front of Cabey.


title: 1984 New York City Subway shooting

text: Whether Goetz actually said aloud the words "You don't look so bad, here's another" is still a matter of dispute. He has subsequently denied on several occasions making the statement. One source said, "In all probability, the defendant uttered these words only to himself and probably not even mouthing the words, but just saying them in his own mind as he squeezed the trigger that fifth time."


title: 1984 New York City Subway shooting

text: The terrified passengers ran to the other end and out of the car, leaving behind two women who had been close to the shooting, immobilized by fear. According to Goetz's statements to Concord police department, Canty was lying on the subway floor to the right of a seated woman who had remained in the same position she had been when previously seated near Canty. This woman was staring straight ahead in shock. Allen was laying on his stomach on her bench with his head in her lap. Ramseur was slumped on the floor near the north end door of the subway car, and Cabey was slumped on the floor near him next to the conductor's cab. A second woman near the center of the subway car had fainted and was lying face down. The male who prior to the shooting was sitting on Goetz's left had exited or was in the process of exiting the car. Goetz asked the two frightened women if they had been injured and both said they were not. He then went towards the south end of the car, briefly looked around, and sat down with his head in his hands covering his eyes, occasionally looking around. Shortly afterwards the train conductor entered the car and loudly exclaimed "What's going on?" The conductor approached Goetz, still seated, and asked him what happened. Goetz pointed to the north end of the car and then told him, "I don't know ... they tried to rob me and I shot them." The conductor then went to the two women to check if they were injured. The conductor returned to Goetz and asked if he was a police officer, receiving the reply, "No." and then asked Goetz for the gun, receiving the reply, "No." The conductor then left the car.


title: 1984 New York City Subway shooting

text: Some time later the train slowed down, continued at a slow rate, and then stopped before the next station. Goetz remained seated for 15–30 seconds, and then exited the south end of the car. He climbed down to the tracks, ran north a few steps, and then reversed direction and ran south through the tunnel to the Chambers Street station, where he climbed onto the empty end of the passenger platform and then exited the station via the stairs. Later Goetz said sirens were going off everywhere. He took a cab back home to 14th Street to gather some belongings, and then rented a car and drove north to Bennington, Vermont, where he burned the blue jacket he had been wearing and dismantled the revolver, scattering the pieces in the woods north of town. He drove around New England for several days, registering at motels under various names and paying in cash.


title: 1984 New York City Subway shooting

text: On December 26, an anonymous hotline caller told New York City police that Goetz matched the gunman's description, owned a gun, and had been mugged previously. On December 29, Goetz called his neighbor, Myra Friedman, who told him that police had come by his apartment looking for him, and had left notes asking to be contacted as soon as possible. He gave his side of the story to Friedman, and described his psychological state at the time:


title: 1984 New York City Subway shooting

text: Goetz returned to New York City on December 30, turned in the car, picked up some clothing and business papers at his apartment, rented another car and drove back to New England. Shortly after noon the next day, he walked into the Concord, New Hampshire, police headquarters and told the officer on duty, "I am the person they are seeking in New York."


title: 1984 New York City Subway shooting

text: Once the officer realized that Goetz was a genuine suspect, Goetz was given a Miranda warning and he waived his right to have an attorney present. After an interview that lasted over an hour, a Concord detective asked Goetz to consent to making an audiotaped statement. Goetz agreed, and a two-hour statement was recorded. That evening, New York City detectives and an assistant district attorney arrived in Concord, and Goetz submitted to a two-hour videotaped interview. Both interviews were eventually played back for the grand juries, the criminal trial, and a civil trial years later. When the audiotape was first played in open court, Goetz was described by "The New York Times" as "confused and emotional, alternately horrified by and defensive about his actions, and obsessed with justifying them."


title: 1984 New York City Subway shooting

text: In his statements, Goetz described his past mugging, in which he was injured and the only assailant arrested went unpunished. He called New York City "lawless" and expressed contempt for its justice system, calling it a "joke," a "sham," and "a disgrace". Goetz said that when the four surrounded him on the train, he feared being "beaten to a pulp" as well as being robbed. He denied any premeditation for the shooting, something that had been speculated on by the press. Asked what his intentions were when he drew his revolver, Goetz replied, "My intention was to murder them, to hurt them, to make them suffer as much as possible." Later in the tape, Goetz said, "If I had more bullets, I would have shot 'em all again and again. My problem was I ran out of bullets." He added, "I was gonna, I was gonna gouge one of the guys' [Canty's] eyes out with my keys afterwards", but said he stopped when he saw the fear in his eyes. At the criminal trial, Goetz's defense attorneys, Barry Slotnick and Mark M. Baker, argued that this and other extreme statements by Goetz were the product of emotion and an overactive imagination.