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Quantum field theory
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Quantum field theory (QFT)[1] provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or (especially in a condensed matter context) many-body systems. It is widely used in particle physics and condensed matter physics. Most theories in modern particle physics, including the Standard Model of elementary particles and their interactions, are formulated as relativistic quantum field theories. Quantum field theories are used in many circumstances, especially those where the number of particles fluctuates—for example, in the BCS theory of superconductivity.

In perturbative quantum field theory, the forces between particles are mediated by other particles. The electromagnetic force between two electrons is caused by an exchange of photons. Intermediate vector bosons mediate the weak force and gluons mediate the strong force. There is currently no complete quantum theory of the remaining fundamental force, gravity, but many of the proposed theories postulate the existence of a graviton particle which mediates it. These force-carrying particles are virtual particles and, by definition, cannot be detected while carrying the force, because such detection will imply that the force is not being carried. In addition, the notion of "force mediating particle" comes from perturbation theory, and thus does not make sense in a context of bound states.

In QFT photons are not thought of as 'little billiard balls', they are considered to be field quanta - necessarily chunked ripples in a field that 'look like' particles. Fermions, like the electron, can also be described as ripples in a field, where each kind of fermion has its own field. In summary, the classical visualisation of "everything is particles and fields", in quantum field theory, resolves into "everything is particles", which then resolves into "everything is fields". In the end, particles are regarded as excited states of a field (field quanta).



Quantum field theory originated in the 1920s from the problem of creating a quantum mechanical theory of the electromagnetic field. In 1926, Max Born, Pascual Jordan, and Werner Heisenberg constructed such a theory by expressing the field's internal degrees of freedom as an infinite set of harmonic oscillators and by employing the usual procedure for quantizing those oscillators (canonical quantization). Max Plank, a physicist at the University of Kiels, observed the behavior at the atomic level of radiation and heat on matter. He observed that the energy absorbed or emitted was contained in small, discrete (i.e. individual) energy packets called quanta. This theory assumed that no electric charges or currents were present, and today would be called a free field theory. The first reasonably complete theory of quantum electrodynamics, which included both the electromagnetic field and electrically charged matter (specifically, electrons) as quantum mechanical objects, was created by Paul Dirac in 1927[2]. This quantum field theory could be used to model important processes such as the emission of a photon by an electron dropping into a quantum state of lower energy, a process in which the number of particles changes — one atom in the initial state becomes an atom plus a photon in the final state. It is now understood that the ability to describe such processes is one of the most important features of quantum field theory.

It was evident from the beginning that a proper quantum treatment of the electromagnetic field had to somehow incorporate Einstein's relativity theory, which had after all grown out of the study of classical electromagnetism. This need to put together relativity and quantum mechanics was the second major motivation in the development of quantum field theory. Pascual Jordan and Wolfgang Pauli showed in 1928 that quantum fields could be made to behave in the way predicted by special relativity during coordinate transformations (specifically, they showed that the field commutators were Lorentz invariant), and in 1933 Niels Bohr and Leon Rosenfeld showed that this result could be interpreted as a limitation on the ability to measure fields at space-like separations, exactly as required by relativity. A further boost for quantum field theory came with the discovery of the Dirac equation, a single-particle equation obeying both relativity and quantum mechanics, when it was shown that several of its undesirable properties (such as negative-energy states) could be eliminated by reformulating the Dirac equation as a quantum field theory. This work was performed by Wendell Furry, Robert Oppenheimer, Vladimir Fock, and others.

The third thread in the development of quantum field theory was the need to handle the statistics of many-particle systems consistently and with ease. In 1927, Jordan tried to extend the canonical quantization of fields to the many-body wavefunctions of identical particles, a procedure that is sometimes called second quantization. In 1928, Jordan and Eugene Wigner found that the quantum field describing electrons, or other fermions, had to be expanded using anti-commuting creation and annihilation operators due to the Pauli exclusion principle. This thread of development was incorporated into many-body theory, and strongly influenced condensed matter physics and nuclear physics.

Despite its early successes, quantum field theory was plagued by several serious theoretical difficulties. Many seemingly-innocuous physical quantities, such as the energy shift of electron states due to the presence of the electromagnetic field, gave infinity — a nonsensical result — when computed using quantum field theory. This "divergence problem" was solved during the 1940s by Bethe, Tomonaga, Schwinger, Feynman, and Dyson, through the procedure known as renormalization. This phase of development culminated with the construction of the modern theory of quantum electrodynamics (QED). Beginning in the 1950s with the work of Yang and Mills, QED was generalized to a class of quantum field theories known as gauge theories. The 1960s and 1970s saw the formulation of a gauge theory now known as the Standard Model of particle physics, which describes all known elementary particles and the interactions between them. The weak interaction part of the standard model was formulated by Sheldon Glashow, with the Higgs mechanism added by Steven Weinberg and Abdus Salam. The theory was shown to be renormalizable and hence consistent by Gerardus 't Hooft and Martinus Veltman.

Also during the 1970s, parallel developments in the study of phase transitions in condensed matter physics led Leo Kadanoff, Michael Fisher and Kenneth Wilson (extending work of Ernst Stueckelberg, Andre Peterman, Murray Gell-Mann and Francis Low) to a set of ideas and methods known as the renormalization group. By providing a better physical understanding of the renormalization procedure invented in the 1940s, the renormalization group sparked what has been called the "grand synthesis" of theoretical physics, uniting the quantum field theoretical techniques used in particle physics and condensed matter physics into a single theoretical framework.


Principles of quantum field theory


Classical fields and quantum fields

Quantum mechanics, in its most general formulation, is a theory of abstract operators (observables) acting on an abstract state space (Hilbert space), where the observables represent physically-observable quantities and the state space represents the possible states of the system under study. Furthermore, each observable corresponds, in a technical sense, to the classical idea of a degree of freedom. For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the position and momentum operators \hat{x} and \hat{p}. Ordinary quantum mechanics deals with systems such as this, which possess a small set of degrees of freedom.

(It is important to note, at this point, that this article does not use the word "particle" in the context of wave–particle duality. In quantum field theory, "particle" is a generic term for any discrete quantum mechanical entity, such as an electron or photon, which can behave like classical particles or classical waves under different experimental conditions.)

A quantum field is a quantum mechanical system containing a large, and possibly infinite, number of degrees of freedom. This is not as exotic a situation as one might think (e.g., in an infinite-dimensional vector space, a vector is just a function as ordinary as f(x) = x^2; which is, of course, familiar territory). A classical field contains a set of degrees of freedom at each point of space; for instance, the classical electromagnetic field defines two vectors — the electric field and the magnetic field — that can in principle take on distinct values for each position r. When the field as a whole is considered as a quantum mechanical system, its observables form an infinite (in fact uncountable) set, because r is continuous.

Furthermore, the degrees of freedom in a quantum field are arranged in "repeated" sets. For example, the degrees of freedom in an electromagnetic field can be grouped according to the position r, with exactly two vectors for each r. Note that r is an ordinary number that "indexes" the observables; it is not to be confused with the position operator \hat{x} encountered in ordinary quantum mechanics, which is an observable. (Thus, ordinary quantum mechanics is sometimes referred to as "zero-dimensional quantum field theory", because it contains only a single set of observables.)

It is also important to note that there is nothing special about r because, as it turns out, there is generally more than one way of indexing the degrees of freedom in the field.

In the following sections, we will show how these ideas can be used to construct a quantum mechanical theory with the desired properties. We will begin by discussing single-particle quantum mechanics and the associated theory of many-particle quantum mechanics. Then, by finding a way to index the degrees of freedom in the many-particle problem, we will construct a quantum field and study its implications.

Single-particle and many-particle quantum mechanics

In ordinary quantum mechanics, the time-dependent one-dimensional Schrödinger equation describing the time evolution of the quantum state of a single non-relativistic particle is

 \left[ \frac{-{\hbar}^2}{2m}\frac{{\partial}^2}{\partial x^2} + V(\mathbf{x}) \right] |\psi(t)\rang = i \hbar \frac{\partial}{\partial t} |\psi(t)\rang,

where m is the particle's mass, V is the applied potential, and |\psi\rang denotes the quantum state (we are using bra-ket notation).

We wish to consider how this problem generalizes to N particles. There are two motivations for studying the many-particle problem. The first is a straightforward need in condensed matter physics, where typically the number of particles is on the order of Avogadro's number (6.0221415 x 1023). The second motivation for the many-particle problem arises from particle physics and the desire to incorporate the effects of special relativity. If one attempts to include the relativistic rest energy into the above equation (in quantum mechanics where position is an observable), the result is either the Klein-Gordon equation or the Dirac equation. However, these equations have many unsatisfactory qualities; for instance, they possess energy eigenvalues which extend to –∞, so that there seems to be no easy definition of a ground state. It turns out that such inconsistencies arise from relativistic wavefunctions having a probabilistic interpretation in position space, as probability conservation is not a relativistically covariant concept. In quantum field theory, unlike in quantum mechanics, position is not an observable, and thus, one does not need the concept of a position-space probability density. For quantum fields whose interaction can be treated perturbatively, this is equivalent to neglecting the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativistic quantum theory. Einstein's famous mass-energy relation allows for the possibility that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles. For example, an electron and a positron can annihilate each other to create photons. This suggests that a consistent relativistic quantum theory should be able to describe many-particle dynamics.

Furthermore, we will assume that the N particles are indistinguishable. As described in the article on identical particles, this implies that the state of the entire system must be either symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. These multi-particle states are rather complicated to write. For example, the general quantum state of a system of N bosons is written as

 |\phi_1 \cdots \phi_N \rang = \sqrt{\frac{\prod_j N_j!}{N!}} \sum_{p\in S_N} |\phi_{p(1)}\rang \cdots |\phi_{p(N)} \rang,

where |\phi_i\rang are the single-particle states, Nj is the number of particles occupying state j, and the sum is taken over all possible permutations p acting on N elements. In general, this is a sum of N! (N factorial) distinct terms, which quickly becomes unmanageable as N increases. The way to simplify this problem is to turn it into a quantum field theory.

Second quantization

In this section, we will describe a method for constructing a quantum field theory called second quantization. This basically involves choosing a way to index the quantum mechanical degrees of freedom in the space of multiple identical-particle states. It is based on the Hamiltonian formulation of quantum mechanics; several other approaches exist, such as the Feynman path integral[3], which uses a Lagrangian formulation. For an overview, see the article on quantization.

Second quantization of bosons


 \frac{1}{\sqrt{3}} \left[ |\phi_1\rang |\phi_2\rang |\phi_2\rang + |\phi_2\rang |\phi_1\rang |\phi_2\rang + |\phi_2\rang |\phi_2\rang |\phi_1\rang \right].


 |1, 2, 0, 0, 0, \cdots \rangle.

The next step is to expand the N-particle state space to include the state spaces for all possible values of N. This extended state space, known as a Fock space, is composed of the state space of a system with no particles (the so-called vacuum state), plus the state space of a 1-particle system, plus the state space of a 2-particle system, and so forth. It is easy to see that there is a one-to-one correspondence between the occupation number representation and valid boson states in the Fock space.

At this point, the quantum mechanical system has become a quantum field in the sense we described above. The field's elementary degrees of freedom are the occupation numbers, and each occupation number is indexed by a number j\cdots, indicating which of the single-particle states |\phi_1\rang, |\phi_2\rang, \cdots|\phi_j\rang\cdots it refers to.

The properties of this quantum field can be explored by defining creation and annihilation operators, which add and subtract particles. They are analogous to "ladder operators" in the quantum harmonic oscillator problem, which added and subtracted energy quanta. However, these operators literally create and annihilate particles of a given quantum state. The bosonic annihilation operator a2 and creation operator a_2^\dagger have the following effects:

 a_2 | N_1, N_2, N_3, \cdots \rangle = \sqrt{N_2} \mid N_1, (N_2 - 1), N_3, \cdots \rangle,
 a_2^\dagger | N_1, N_2, N_3, \cdots \rangle = \sqrt{N_2 + 1} \mid N_1, (N_2 + 1), N_3, \cdots \rangle.


 \left[a_i , a_j \right] = 0 \quad,\quad \left[a_i^\dagger , a_j^\dagger \right] = 0 \quad,\quad \left[a_i , a_j^\dagger \right] = \delta_{ij},


The Hamiltonian of the quantum field (which, through the Schrödinger equation, determines its dynamics) can be written in terms of creation and annihilation operators. For instance, the Hamiltonian of a field of free (non-interacting) bosons is

H = \sum_k E_k \, a^\dagger_k \,a_k,

where Ek is the energy of the k-th single-particle energy eigenstate. Note that

a_k^\dagger\,a_k|\cdots, N_k, \cdots \rangle=N_k| \cdots, N_k, \cdots \rangle.

Second quantization of fermions


 c_j | N_1, N_2, \cdots, N_j = 0, \cdots \rangle = 0
 c_j | N_1, N_2, \cdots, N_j = 1, \cdots \rangle = (-1)^{(N_1 + \cdots + N_{j-1})} | N_1, N_2, \cdots, N_j = 0, \cdots \rangle
 c_j^\dagger | N_1, N_2, \cdots, N_j = 0, \cdots \rangle = (-1)^{(N_1 + \cdots + N_{j-1})} | N_1, N_2, \cdots, N_j = 1, \cdots \rangle
 c_j^\dagger | N_1, N_2, \cdots, N_j = 1, \cdots \rangle = 0.

These obey an anticommutation relation:

 \left\{c_i , c_j \right\} = 0 \quad,\quad \left\{c_i^\dagger , c_j^\dagger \right\} = 0 \quad,\quad \left\{c_i , c_j^\dagger \right\} = \delta_{ij}.


Field operators





The bosonic field operators obey the commutation relation

 \left[\phi(\mathbf{r}) , \phi(\mathbf{r'}) \right] = 0 \quad,\quad \left[\phi^\dagger(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = 0 \quad,\quad \left[\phi(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = \delta^3(\mathbf{r} - \mathbf{r'})


It should be emphasized that the field operator is not the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is just a scalar field. However, they are closely related, and are indeed commonly denoted with the same symbol. If we have a Hamiltonian with a space representation, say


where the indices i and j run over all particles, then the field theory Hamiltonian is


This looks remarkably like an expression for the expectation value of the energy, with φ playing the role of the wavefunction. This relationship between the field operators and wavefunctions makes it very easy to formulate field theories starting from space-projected Hamiltonians.

Implications of quantum field theory

Unification of fields and particles



Physical meaning of particle indistinguishability



Particle conservation and non-conservation


For example, we can see that the free-boson Hamiltonian described above conserves particle number. Whenever the Hamiltonian operates on a state, each particle destroyed by an annihilation operator ak is immediately put back by the creation operator a_k^\dagger.





Axiomatic approaches




Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. One of the Millennium Prize Problems—proving the existence of a mass gap in Yang-Mills theory—is linked to this issue.

Phenomena associated with quantum field theory

In the previous part of the article, we described the most general properties of quantum field theories. Some of the quantum field theories studied in various fields of theoretical physics possess additional special properties, such as renormalizability, gauge symmetry, and supersymmetry. These are described in the following sections.



Many of these problems are related to failures in classical electrodynamics that were identified but unsolved in the 19th century, and they basically stem from the fact that many of the supposedly "intrinsic" properties of an electron are tied to the electromagnetic field which it carries around with it. The energy carried by a single electron—its self energy—is not simply the bare value, but also includes the energy contained in its electromagnetic field, its attendant cloud of photons. The energy in a field of a spherical source diverges in both classical and quantum mechanics, but as discovered by Weisskopf, in quantum mechanics the divergence is much milder, going only as the logarithm of the radius of the sphere.





Gauge freedom




In general, the gauge transformations of a theory consist of several different transformations, which may not be commutative. These transformations are together described by a mathematical object known as a gauge group. Infinitesimal gauge transformations are the gauge group generators. Therefore the number of gauge bosons is the group dimension (i.e. number of generators forming a basis).






See also


  1. ^ Weinberg, S. Quantum Field Theory, Vols. I to III, 2000, Cambridge University Press: Cambridge, UK.
  3. ^ Abraham Pais, Inward Bound: Of Matter and Forces in the Physical World ISBN 0198519974. Pais recounts how his astonishment at the rapidity with which Feynman could calculate using his method. Feynman's method is now part of the standard methods for physicists.

Further reading

General readers:

Introductory texts:

Advanced texts:


External links


Up to date as of January 23, 2010


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