The alternative to the use of fictitious imaginary times is to use a real-time formalism which come in two forms. A path-ordered approach to real-time formalisms includes the Schwinger–Keldysh formalism and more modern variants. The latter involves replacing a straight time contour from (large negative) real initial time t i to t i − − i β β by one that first runs to (large positive) real time t f and then suitably back to t i − − i β β . In fact all that is needed is one section running along the real time axis, as the route to the end point, t i − − i β β , is less important. The piecewise composition of the resulting complex time contour leads to a doubling of fields and more complicated Feynman rules, but obviates the need of analytic continuations of the imaginary-time formalism. The alternative approach to real-time formalisms is an operator based approach using Bogoliubov transformations, known as thermo field dynamics. As well as Feynman diagrams and perturbation theory, other techniques such as dispersion relations and the finite temperature analog of Cutkosky rules can also be used in the real time formulation.