The path integral itself also deals with larger mathematical spaces than is usual, which requires more careful mathematics, not all of which has been fully worked out. The path integral historically was not immediately accepted, partly because it took many years to incorporate fermions properly.
This required physicists to invent an entirely new mathematical object — the Grassmann variable — which also allowed changes of variables to be done naturally, as well as allowing constrained quantization. The integration variables in the path integral are subtly non-commuting. The value of the product of two field operators at what looks like the same point depends on how the two points are ordered in space and time.
This makes some naive identities fail.
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In relativistic theories, there is both a particle and field representation for every theory. The field representation is a sum over all field configurations, and the particle representation is a sum over different particle paths. The nonrelativistic formulation is traditionally given in terms of particle paths, not fields. There, the path integral in the usual variables, with fixed boundary conditions, gives the probability amplitude for a particle to go from point x to point y in time T :. This is called the propagator. The Fourier transform in time, extending K p ; T to be zero for negative times, gives Green's function, or the frequency-space propagator:.
The infinitesimal term in the denominator is a small positive number, which guarantees that the inverse Fourier transform in E will be nonzero only for future times. For past times, the inverse Fourier transform contour closes toward values of E where there is no singularity.
This guarantees that K propagates the particle into the future and is the reason for the subscript "F" on G. The infinitesimal term can be interpreted as an infinitesimal rotation toward imaginary time. In this case, the interpretation is that these are the quantities to convolve the final wavefunction so as to get the initial wavefunction:. For a nonrelativistic theory, the time as measured along the path of a moving particle and the time as measured by an outside observer are the same. In relativity, this is no longer true.
For a relativistic theory the propagator should be defined as the sum over all paths that travel between two points in a fixed proper time, as measured along the path these paths describe the trajectory of a particle in space and in time :. The integral above is not trivial to interpret because of the square root.
Fortunately, there is a heuristic trick. The sum is over the relativistic arc length of the path of an oscillating quantity, and like the nonrelativistic path integral should be interpreted as slightly rotated into imaginary time. This can be given a probability interpretation. The sum over all paths is a probability average over a path constructed step by step.
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The usual definition of the relativistic propagator only asks for the amplitude is to travel from x to y , after summing over all the possible proper times it could take:. This is the Schwinger representation.
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So in p -space, the propagator can be reexpressed simply:. This expression can be interpreted in the nonrelativistic limit, where it is convenient to split it by partial fractions :. When convolving with the propagator, which in p space just means multiplying by the propagator, the second term is suppressed and the first term is enhanced. The second term has a nonrelativistic limit also, but this limit is concentrated on frequencies that are negative.
The second pole is dominated by contributions from paths where the proper time and the coordinate time are ticking in an opposite sense, which means that the second term is to be interpreted as the antiparticle.
The nonrelativistic analysis shows that with this form the antiparticle still has positive energy. In the Fourier transform, this means shifting the pole in p 0 slightly, so that the inverse Fourier transform will pick up a small decay factor in one of the time directions:. Without these terms, the pole contribution could not be unambiguously evaluated when taking the inverse Fourier transform of p 0.
Gert Roepstorff (Author of Path Integral Approach to Quantum Physics)
The terms can be recombined:. This is the mathematically precise form of the relativistic particle propagator, free of any ambiguities. So in the relativistic case, the Feynman path-integral representation of the propagator includes paths going backwards in time, which describe antiparticles. The paths that contribute to the relativistic propagator go forward and backwards in time, and the interpretation of this is that the amplitude for a free particle to travel between two points includes amplitudes for the particle to fluctuate into an antiparticle, travel back in time, then forward again.
Unlike the nonrelativistic case, it is impossible to produce a relativistic theory of local particle propagation without including antiparticles. All local differential operators have inverses that are nonzero outside the light cone, meaning that it is impossible to keep a particle from travelling faster than light. Such a particle cannot have a Green's function which is only nonzero in the future in a relativistically invariant theory.
However, the path integral formulation is also extremely important in direct application to quantum field theory, in which the "paths" or histories being considered are not the motions of a single particle, but the possible time evolutions of a field over all space. In principle, one integrates Feynman's amplitude over the class of all possible field configurations.
Much of the formal study of QFT is devoted to the properties of the resulting functional integral, and much effort not yet entirely successful has been made toward making these functional integrals mathematically precise. Such a functional integral is extremely similar to the partition function in statistical mechanics. Indeed, it is sometimes called a partition function , and the two are essentially mathematically identical except for the factor of i in the exponent in Feynman's postulate 3.
Analytically continuing the integral to an imaginary time variable called a Wick rotation makes the functional integral even more like a statistical partition function and also tames some of the mathematical difficulties of working with these integrals. As stated above, the unadorned path integral in the denominator ensures proper normalization.
Strictly speaking, the only question that can be asked in physics is: What fraction of states satisfying condition A also satisfy condition B? The answer to this is a number between 0 and 1, which can be interpreted as a conditional probability , written as P B A. In particular, this could be a state corresponding to the state of the Universe just after the Big Bang , although for actual calculation this can be simplified using heuristic methods.
Since this expression is a quotient of path integrals, it is naturally normalised. Since this formulation of quantum mechanics is analogous to classical action principle, one might expect that identities concerning the action in classical mechanics would have quantum counterparts derivable from a functional integral. This is often the case. In the language of functional analysis, we can write the Euler—Lagrange equations as. The quantum analogues of these equations are called the Schwinger—Dyson equations. In the deWitt notation this looks like .
These equations are the analog of the on-shell EL equations.
The time ordering is taken before the time derivatives inside the S , i. If J called the source field is an element of the dual space of the field configurations which has at least an affine structure because of the assumption of the translational invariance for the functional measure , then the generating functional Z of the source fields is defined to be. For example, if.
Then, from the properties of the functional integrals. This is true, for example, for nonlinear sigma models where the target space is diffeomorphic to R n. However, if the target manifold is some topologically nontrivial space, the concept of a translation does not even make any sense. In that case, we would have to replace the S in this equation by another functional. The path integrals are usually thought of as being the sum of all paths through an infinite space—time.
However, in local quantum field theory we would restrict everything to lie within a finite causally complete region, for example inside a double light-cone. This gives a more mathematically precise and physically rigorous definition of quantum field theory. Now how about the on shell Noether's theorem for the classical case? Does it have a quantum analog as well? Yes, but with a caveat.
Introduction to quantum mechanics: Schrodinger equation and path integral
The functional measure would have to be invariant under the one parameter group of symmetry transformation as well. Let's just assume for simplicity here that the symmetry in question is local not local in the sense of a gauge symmetry , but in the sense that the transformed value of the field at any given point under an infinitesimal transformation would only depend on the field configuration over an arbitrarily small neighborhood of the point in question.
Let's also assume that the action is local in the sense that it is the integral over spacetime of a Lagrangian , and that. Here, Q is a derivation which generates the one parameter group in question. We could have antiderivations as well, such as BRST and supersymmetry. This property is called the invariance of the measure. And this does not hold in general.
See anomaly physics for more details. Now, let's assume even further that Q is a local integral. More general Lagrangians would require a modification to this definition! We're not insisting that q x is the generator of a symmetry i. And we also assume the even stronger assumption that the functional measure is locally invariant:.