Path integral formulation

This article is about a formulation of quantum mechanics. For integrals along a path, also known as line or contour integrals, see Path integral.

The path integral formulation of quantum mechanics was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier, in the course of his doctoral thesis work with John Archibald Wheeler. It is a description of quantum theory which generalizes 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.

This formulation has proved crucial to the subsequent development of theoretical physics, since it provided the basis for the grand synthesis of the 1970's called the renormalization group which unified quantum field theory with statistical mechanics. It is no surprise, therefore, that path integrals have also been used in the study of Brownian motion and diffusion.

Formulating quantum mechanics

The path integral method is an alternative formulation of quantum mechanics. The canonical approach, pioneered by Schroedinger, Heisenberg and Dirac paid great attention to wave-particle duality and the resulting uncertainty principle by replacing Poisson brackets of classical mechanics by commutators between operators in quantum mechanics. The Hilbert space of quantum states and the superposition law of quantum amplitudes follows. The path integral starts from the superposition law, and exploits wave-particle duality to build a generating function for quantum amplitudes.

Quantum amplitudes

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 <math> e^{\frac{i}{\hbar}I(H)}<math>, 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 adds up, or 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. Not only that, it assigns all of them, no matter how bizarre, amplitudes of equal size; only the phase, or argument of the complex number, varies. The contributions wildly different from the classical history are suppressed only by the interference of similar histories (see below).

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.

The equal magnitude of all amplitudes in the path integral tends to make it difficult to define it such that it converges and is mathematically tractable. For purposes of actual evaluation of quantities using path-integral methods, it is common to give the action an imaginary part in order to damp the wilder contributions to the integral, then take the limit of a real action at the end of the calculation. In quantum field theory this takes the form of Wick rotation.

There is some difficulty in defining a measure over the space of paths. In particular, the measure is concentrated on "fractallike" distributional paths.

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 <math>\hbar<math>, 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 position <math>x_0<math> at time <math>0<math> to <math>x_n<math> at time <math>t<math>, the time interval can be divided up into little segments of duration <math>\Delta t<math> and the path integral can be computed as proportional to

<math>

\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))} <math>

where <math>H<math> is the entire history in which the particle zigzags from its initial to its final position linearly between all the values of

<math>x_j = x(j \Delta t)<math>.

In the limit, this is an integral over an infinite-dimensional space—a functional integral.

The path integral and the partition function

The path integral is just the generalization of the integral above to all quantum mechanical problems—

<math>Z = \int Dx e^{iS[x]/\hbar}<math>  where  <math>S[x]=\int_0^T dt L[x(t)]<math>

is the action of the classical problem in which one investigates the path starting at time t=0 and ending at time t=T, and Dx denotes integration over all paths. In the classical limit, <math>\hbar\to0<math>, the path of minimum action dominates the integral, because the phase of any path away from this fluctuates rapidly and different contributions cancel.

The connection with statistical mechanics follows. Perform the Wick rotation t→it, ie, make time imaginary. Then the path integral resembles the partition function of statistical mechanics defined in a canonical ensemble with temperature <math>1/T\hbar<math>.

Clearly, such a deep analogy between quantum mechanics and statistical mechanics cannot be dependent on the formulation. In the canonical formulation, one sees that the unitary evolution operator of a state is given by

<math>|\alpha;t\rangle=e^{iHt\hbar}|\alpha;0\rangle<math>

where the state α is evolved from time t=0. If one makes a Wick rotation here, and finds the amplitude to go from any state, back to the same state in (imaginary) time iT is given by

<math>Z={\rm Tr} e^{-HT\hbar}<math>

which is precisely the partition function of statistical mechanics for the same system at temperature quoted earlier. One aspect of this equivalence was also known to Schroedinger who remarked that the equation named after him looked like the diffusion equation after Wick rotation.

Quantum field theory

Today, the most common use of the path-integral formulation is in quantum field theory.

The propagator

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. One way to do this, which Feynman used to explain photon and electron/positron propagators in quantum electrodynamics, is to apply the path integral to the motion of a single particle—one, however, that can roam back and forth through time as well as space in the course of its wanderings. (Such behavior can be reinterpreted as the contribution of the creation and annihilation of virtual particle-antiparticle pairs, so in this sense the single-particle restriction has already been loosened.)

Functionals of fields

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: <math>S[\phi]<math> where the field <math>\phi (x^\mu)<math> is itself a function of space and time, and the square brackets are a reminder that the action depends on all the field's values everywhere, not just some particular value. 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 <math>i<math> 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.

Expectation values

In quantum field theory, if the action is given by the functional S of field configurations (which only depends locally on the fields), then the time ordered vacuum expectation value of polynomially bounded functional F, <F>, is given by

<math>\left\langle F\right\rangle=\frac{\int \mathcal{D}\phi F[\phi]e^{iS[\phi]}}{\int\mathcal{D}\phi e^{iS[\phi]}}<math>

The symbol <math>\int \mathcal{D}\phi<math> here is a concise way to represent the infinite-dimensional integral over all possible field configurations on all of space-time. As stated above, we put the unadorned path integral in the denominator to normalize everything properly.

Schwinger-Dyson equations

Since this formulation of quantum mechanics is analogous to classical action principles, 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 <math>\frac{\delta}{\delta \phi}S[\phi]=0<math> (the left-hand side is a functional derivative; the equation means that the action is stationary under small changes in the field configuration). The quantum analogues of these equations are called the Schwinger-Dyson equations.

If the functional measure <math>\mathcal{D}\phi<math> turns out to be translationally invariant (we'll assume this for the rest of this article, although this does not hold for, let's say nonlinear sigma models) and if we assume that after a Wick rotation

<math>e^{iS[\phi]}<math>,

which now becomes

<math>e^{-H[\phi]}<math>

for some H, goes to zero faster than any reciprocal of any polynomial for large values of φ, we can integrate by parts (after a Wick rotation, followed by a Wick rotation back) to get the following Schwinger-Dyson equations:

<math>\left\langle \frac{\delta}{\delta \phi}F[\phi] \right\rangle = -i \left\langle F[\phi]\frac{\delta}{\delta\phi}S[\phi] \right\rangle<math>

for any polynomially bounded functional F.

<math>\left\langle F_{,i} \right\rangle = -i \left\langle F S_{,i} \right\rangle<math>

in the deWitt notation.

These equations are the analog of the on shell EL equations.

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:

<math>Z[J]=\int \mathcal{D}\phi e^{i(S[\phi] + \left\langle J,\phi \right\rangle)}<math>

Note that

<math>\frac{\delta^n Z}{\delta J(x_1) \cdots \delta J(x_n)}[J] = i^n \, Z[J] \, {\left\langle \phi(x_1)\cdots \phi(x_n)\right\rangle}_J<math>

or

<math>Z^{,i_1\dots i_n}[J]=i^n Z[J] {\left \langle \phi^{i_1}\cdots \phi^{i_n}\right\rangle}_J<math>

where

<math>{\left\langle F \right\rangle}_J=\frac{\int \mathcal{D}\phi F[\phi]e^{i(S[\phi] + \left\langle J,\phi \right\rangle)}}{\int\mathcal{D}\phi e^{i(S[\phi] + \left\langle J,\phi \right\rangle)}}<math>

Basically, if <math>\mathcal{D}\phi e^{iS[\phi]}<math> is viewed as a functional distribution (this shouldn't be taken too literally as an interpretation of QFT, unlike its Wick rotated statistical mechanics analogue, because we have time ordering complications here!), then <math>\left\langle\phi(x_1)\cdots \phi(x_n)\right\rangle<math> are its moments and Z is its Fourier transform.

If F is a functional of φ, then for an operator K, F[K] is defined to be the operator which substitutes K for φ. For example, if

<math>F[\phi]=\frac{\partial^{k_1}}{\partial x_1^{k_1}}\phi(x_1)\cdots \frac{\partial^{k_n}}{\partial x_n^{k_n}}\phi(x_n)<math>

and G is a functional of J, then

<math>F\left[-i\frac{\delta}{\delta J}\right] G[J] = (-i)^n \frac{\partial^{k_1}}{\partial x_1^{k_1}}\frac{\delta}{\delta J(x_1)} \cdots \frac{\partial^{k_n}}{\partial x_n^{k_n}}\frac{\delta}{\delta J(x_n)} G[J]<math>.

Then, from the properties of the functional integrals, we get the "master" Schwinger-Dyson equation:

<math>\frac{\delta S}{\delta \phi(x)}\left[-i \frac{\delta}{\delta J}\right]Z[J]+J(x)Z[J]=0<math>

or

<math>S_{,i}[-i\partial]Z+J_i Z=0<math>

If the functional measure is not translationally invariant, it might be possible to express it as the product <math>M\left[\phi\right]\,\mathcal{D}\phi<math> where M is a functional and <math>\mathcal{D}\phi<math> is a translationally invariant measure. This is true, for example, for nonlinear sigma models where the target space is diffeomorphic to Rⁿ. 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 <math>\hat{S}=S-i\ln(M)<math>

If we expand this equation as a Taylor series about J=0, we get the entire set of Schwinger-Dyson equations.

Ward-Takahashi identities

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. 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 <math>Q[\mathcal{L}(x)]=\partial_\mu f^\mu (x)<math> for some function f where f only depends locally on φ (and possibly the spacetime position).

If we don't assume any special boundary conditions, this would not be a "true" symmetry in the true sense of the term in general unless f=0 or something. Here, Q is a derivation which generates the one parameter group in question. We could have antiderivations as well, such as BRST and supersymmetry.

Let's also assume <math>\int \mathcal{D}\phi Q[F][\phi]=0<math> for any polynomially bounded functional F. This property is called the invariance of the measure. And this does not hold in general. See anomaly (physics) for more details.

Then,

<math>\int \mathcal{D}\phi Q[F e^{iS}][\phi]=0<math>, which implies

<math>\left\langle Q[F]\right\rangle +i\left\langle F\int_{\partial V} f^\mu ds_\mu\right\rangle=0<math>

where the integral is over the boundary. This is the quantum analog of Noether's theorem.

Now, let's assume even further that Q is a local integral <math>Q=\int d^dx q(x)<math> where q(x)[φ(y)]=δ^(d)(x-y)Q[φ(y)] so that <math>q(x)[S]=\partial_\mu j^\mu (x)<math> where <math>j^{\mu}(x)=f^\mu(x)-\frac{\partial}{\partial (\partial_\mu \phi)}\mathcal{L}(x) Q[\phi]<math> (this is assuming the Lagrangian only depends on φ and its first partial derivatives! More general Lagrangians would require a modification to this definition!). Note that we're NOT insisting that q(x) is the generator of a symmetry (i.e. we're NOT insisting upon the gauge principle), but just that Q is. And let's also assume the even stronger assumption that the functional measure is locally invariant:

<math>\int \mathcal{D}\phi q(x)[F][\phi]=0<math>.

Then, we'd have

<math>\left\langle q(x)[F] \right\rangle +i\left\langle F q(x)[S]\right\rangle=\left\langle q(x)[F]\right\rangle +i\left\langle F\partial_\mu j^\mu(x)\right\rangle=0<math>

Alternatively,

<math>q(x)[S][-i \frac{\delta}{\delta J}]Z[J]+J(x)Q[\phi(x)][-i \frac{\delta}{\delta J}]Z[J]=\partial_\mu j^\mu(x)[-i \frac{\delta}{\delta J}]Z[J]+J(x)Q[\phi(x)][-i \frac{\delta}{\delta J}]Z[J]=0<math>

The above two equations are the Ward-Takahashi identities.

Now for the case where f=0, we can forget about all the boundary conditions and locality assumptions. We'd simply have

<math>\left\langle Q[F]\right\rangle =0<math>.

Alternatively,

<math>\int d^dx J(x)Q[\phi(x)][-i \frac{\delta}{\delta J}]Z[J]=0<math>

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. The sum over histories method gives identical results to canonical quantum mechanics, and Sinha and Sorkin claim the interpretation explains the Einstein-Podolsky-Rosen paradox without resorting to nonlocality.

Some advocates of interpretations of quantum mechanics emphasizing decoherence have attempted to make more rigorous the notion of extracting a classical-like "coarse-grained" history from the space of all possible histories.

References

• Feynman, R. P., and Hibbs, A. R., Quantum Physics and Path Integrals, New York: McGraw-Hill, 1965. ISBN 0-070-20650-3
• Glimm, James, and Jaffe, Arthur, Quantum Physics: A Functional Integral Point of View, New York: Springer-Verlag, 1981. ISBN 0-387-90562-6
• Pokorski, Stefan, Gauge Field Theories, Cambridge: Cambridge University Press, 1987. ISBN 0-521-36846-4
• Sakurai, J. J., Modern Quantum Mechanics, Tuan, San Fu, ed. Redwood City, California: Addison-Wesley, 1985. ISBN 0-8053-7501-5
• Sinha, Sukanya and Sorkin, Rafael Dolnick. "A Sum-over-histories Account of an EPR(B) Experiment". Foundations of Physics Letters, 4:303-335, 1991. (also available online: PostScript (http://physics.syr.edu/~sorkin/some.papers/63.eprb.ps))
• Hagen Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition, World Scientific (Singapore, 2004); Paperback ISBN 981-238-107-4 (also available online: PDF-files (http://www.physik.fu-berlin.de/~kleinert/b5))

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