Path integral formulation

Functional integral (QFT)

Richard Feynman developed the path integral formulation of quantum mechanics in 1948 as a description of quantum theory corresponding to the action principle of classical mechanics. It replaces the classical notion of a single, unique history for a system with a sum, or functional integral, over an infinity of possible histories to compute a quantum amplitude.

Postulates

Feynman proposed the following postulates:

1. The probability for any fundamental event is given by the absolute square of a complex amplitude.

2. The amplitude for some event is given by adding together all the histories which include that event.

3. The amplitude a certain history contributes is proportional to $e^{{\frac {i}{\hbar }}I(H)}$, where I(H) is the action of that history, or time integral of the Lagrangian.

In order to find the overall probability amplitude for a given process, then, one integrates the amplitude of postulate 3 over the space of all possible histories of the system in between the initial and final states, including histories that are absurd by classical standards. In calculating the amplitude for a single particle to go from one place to another in a given time, it would be correct to include histories in which the particle describes elaborate curlicues, histories in which the particle shoots off into outer space and flies back again, and so forth. The path integral includes them all.

Feynman showed that his formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics. An amplitude computed according to Feynman's principles will also obey the Schrödinger equation for the Hamiltonian corresponding to the given action.

Feynman's postulates are somewhat ambiguous in that they do not define what an "event" is or the exact proportionality constant in postulate 3. The proportionality problem can be solved by simply normalizing the path integral by dividing the amplitude by the square root of the total probability for something to happen (resulting in that the total probability given by all the normalized amplitudes will be 1, as we would expect). Generally speaking one can simply define the "events" in an operational sense for any given experiment.

Recovering the action principle

Feynman was initially attempting to make sense of a brief remark by Paul Dirac about the quantum equivalent of the action principle in classical mechanics. In the limit of action that is large compared to Planck's constant $\hbar$, the path integral is dominated by solutions which are stationary points of the action, since, there, the amplitudes of similar histories will tend to constructively interfere with one another. Conversely, for paths that are far from being stationary points of the action, the complex phase of the amplitude calculated according to postulate 3 will vary rapidly for similar paths, and amplitudes will tend to cancel. Therefore the important parts of the integral—the significant possibilities—in the limit of large action simply consist of solutions of the Euler-Lagrange equation, and classical mechanics is correctly recovered.

Action principles can seem puzzling to the student of physics because of their seemingly teleological quality: instead of predicting the future from initial conditions, one starts with a combination of initial conditions and final conditions and then finds the path in between, as if the system somehow knows where it's going to go. The path integral is one way of understanding why this works. The system doesn't have to know in advance where it's going; the path integral simply calculates the probability amplitude for a given process, and the stationary points of the action mark neighborhoods of the space of histories for which quantum-mechanical interference will yield large probabilities.

Example: Single-particle mechanics

In the case of the motion of a particle, the path integral can be formally thought of as the small-step limit of an integral over zig-zags: for instance, for one-dimensional motion of a particle from postion $x_{0}$ at time $0$ to $x_{n}$ at time $t$, the time interval can be divided up into little segments of duration $\Delta t$ and the path integral can be computed as proportional to

$\lim _{\Delta t\rightarrow 0}\int _{-\infty }^{+\infty }dx_{1}\int _{-\infty }^{+\infty }dx_{2}\int _{-\infty }^{+\infty }dx_{3}\ldots \int _{-\infty }^{+\infty }dx_{n-1}\ e^{{\frac {i}{\hbar }}I(H(x_{j},t))}$

where $H$ is the entire history in which the particle zigzags from its initial to its final position linearly between all the values of $x_{j}=x(j\Delta t)$.

The path integral in quantum field theory

A common use of the path integral is to calculate <q1,t1|q0,t0>, a quantity (here written in bra-ket notation) known as the propagator. As such it is very useful in quantum field theory, where the propagator is an important component of Feynman diagrams.

In such an application, it is still being applied to the motion of a single particle. 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. The action is referred to technically as a functional of the field. In principle, one integrates Feynman's amplitude over the class of all possible combinations of values that the field could have anywhere in space-time.

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.

The path integral in quantum-mechanical interpretation

In one philosophical interpretation of quantum mechanics, the "sum over histories" interpretation, the path integral is taken to be fundamental and reality is viewed as a single indistinguishable "class" of paths which all share the same events. For this interpretation, it is crucial to understand what exactly an event is. Despite its general unpopularity, the sum over histories method gives identical results to canonical quantum mechanics and also explains the Einstein-Podolsky-Rosen paradox without resorting to nonlocality. This makes it the only form of the theory which can explain this paradox without breaking locality.

References

• Feynman, R. P. , and Hibbs, A. R., Quantum Physics and Path Integrals, New York: McGraw-Hill, 1965.

• Glimm, James, and Jaffe, Arthur, Quantum Physics: A Functional Integral Point of View, New York: Springer-Verlag, 1981.

• Sukanya Sinha and Rafael Dolnick Sorkin. "A Sum-over-histories Account of an EPR(B) Experiment". Foundations of Physics Letters, 4:303-335, 1991. (also available online: PostScript)