The path integral formulation of quantum mechanics 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 1970s which unified quantum field theory with statistical mechanics (see the first three textbooks below). If we realize that the Schrödinger equation is essentially a diffusion equation with an imaginary diffusion constant, then the path integral is a method for summing up all possible of random walks. For this reason path integrals had also been used in the study of Brownian motion and diffusion before they were introduced in quantum mechanics.
The path integral method is an alternative formulation of quantum mechanics. The canonical approach, pioneered by Erwin Schrödinger, Werner Heisenberg and Paul 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.
Feynman proposed the following postulates:
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 assigns all of these histories amplitudes of equal magnitude but with varying phase, or argument of the complex number. The contributions that are wildly different from the classical history are suppressed only by the interference of similar, canceling 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 was extending a classic paper by Paul Dirac on the quantum equivalent of the classical action principle. Dirac knew from earlier semiclassical work that the Maupertuis action, the action-angle variable action, is the classical analog of the phase of the wavefunction in an energy eigenstate. He noted that the transformation between the Maupertuis action and the usual Lagrange action replaces a description of a motion at fixed energy with description of a motion evolving between two fixed times. The phase interpretation of the Maupertuis action implied that the Lagrange action was also a phase, and that it was associated to an infinitesimal path starting and ending at two nearby positions at two nearby times. He noted that the full quantum transition amplitude between two points at two times can be heuristically thought of as a sum of this phase factor over all possible paths, where the phase over each infinitesimal segment is given by the Lagrange action.
Dirac's work was only heuristic because he did not provide a precise prescription to calculate the sum, and he did not show that one could recover the Schrödinger equation or the canonical commutation relations from this rule. This was done by Feynman. Both noted that in the limit of action that is large compared to Planck's constant , the path integral is dominated by solutions which are in the neighbourhood of 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 the 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 (EL equation), and classical mechanics is correctly recovered.
Classical action principles are puzzling because of their seemingly teleological quality: instead of predicting the future from initial conditions, they give a combination of initial conditions and final conditions to find the path in between, as if the system somehow knows where it's going to end up. The path integral explains 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 any given process, and the path goes everywhere. After a long enough time, interference effects guarantee that only the contributions from the stationary points of the action give histories with appreciable probabilities.
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.
For a particle in a smooth potential, the path integral is approximated by Feynman as the small-step limit over zig-zag paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position at time to at time , the time sequence
where is the Lagrangian of the 1d-system with position variable x(t) and velocity considered (see below), and corresponds to the position at the j-th time step, if the time integral is approximated by a sum of n terms.
In the limit of going to infinity, this becomes a functional integral, which - apart from a nonessential factor - is directly the product of the probability amplitudes - more precisely, since one must work with a continuous spectrum, the respective densities - to find the quantum mechanical particle at in the initial state and at in the final state .
Actually is the classical Lagrangian of the one-dimensional system considered, , where is the Hamiltonian, with , and the above-mentioned "zigzagging" corresponds to the appearance of the terms in the Riemannian sum approximating the time integral, which are finally integrated over to with the integration measure is an arbitrary value of the interval corresponding to j, e.g. its center,
Thus in contrast to classical mechanics not only the stationary path contributes but actually all virtual paths between the initial and the final point.
Feynman's time-sliced approximation does not, however, exist for the most important quantum-mechanical path integrals of atoms, due to the singularity of the Coulomb potential at the origin. Only after replacing the time by another path-dependent pseudo-time parameter , the singularity is removed and a time-sliced approximation exists, that is exactly integrable, since it can be made harmonic by a simple coordinate transformation, as discovered in 1979 by H. Duru and Hagen Kleinert . The combination of a path-dependent time transformation and a coordinate transformation is an important tool to solve many path integrals and is called generically (Duru-Kleinert transformation).
The path integral representation gives the quantum amplitude to go from point x to point y as an integral over all paths. For a free particle action ():
the integral can be evaluated explicitly.
To do this, it is conceptually convenient to start without the factor i in the exponential, so that large deviations are suppressed by small numbers, not by cancelling oscillatory contributions.
Splitting the integral into time slices:
where the Dx is interpreted as a finite collection of integrations at each integer multiple of . Each factor in the product is a Gaussian as a function of centered at x(t) with variance . The multiple integrals are a repeated convolution of this Gaussian with copies of itself at adjacent times.
Where the number of convolutions is . The result is easy to evaluate by taking the fourier transform of both sides, so that the convolutions becomes multiplications.
The Fourier transform of the Gaussian G is another Gaussian of reciprocal variance:
and the result is:
The Fourier transform gives K, and it is a Gaussian again with reciprocal variance:
The proportionality constant is not really determined by the time slicing approach, only the ratio of values for different endpoint choices is determined. The proportionality constant should be chosen to ensure that between each two time-slices the time-evolution is quantum-mechanically unitary, but a more illuminating way to fix the normalization is to consider the path integral as a description of a stochastic process.
The result has a probability interpretation. The sum over all paths of the exponential factor can be seen as the sum over each path of the probability of selecting that path. The probability is the product over each segment of the probability of selecting that segment, so that each segment is probabilistically independently chosen. The fact that the answer is a Gaussian spreading linearly in time is the central limit theorem, which can be interpreted as the first historical evaluation of a statistical path integral.
The probability interpretation gives a natural normalization choice. The path integral should be defined so that:
This condition normalizes the Gaussian, and produces a Kernel which obeys the diffusion equation:
For oscillatory path integrals, ones with an i in the numerator, the time-slicing produces convolved Gaussians, just as before. Now, however, the convolution product is marginally singular since it requires careful limits to evaluate the oscillating integrals. To make the factors well defined, the easiest way is to add a small imaginary part to the time increment . Then the same convolution argument as before gives the propagation kernel:
Which, with the same normalization as before (not the sum-squares normalization! this function has a divergent norm), obeys a free Schrödinger equation
This means that any superposition of K's will also obey the same equation, by linearity. Defining
then obeys the free Schrödinger equation just as K does:
The path integral reproduces the Schrödinger equation for the initial and final state even when a potential is present. This is easiest to see by taking a path-integral over infinitesimally separated times.
Since the time separation is infinitesimal and the cancelling oscillations become severe for large values of , the path integral has most weight for y close to x. In this case, to lowest order the potential energy is constant, and only the kinetic energy contribution is nontrivial. The exponential of the action is
The first term rotates the phase of locally by an amount proportional to the potential energy. The second term is the free particle propagator, corresponding to i times a diffusion process. To lowest order in they are additive; in any case one has with (1):
As mentioned, the spread in is diffusive from the free particle propagation, with an extra infinitesimal rotation in phase which slowly varies from point to point from the potential:
and this is the Schrödinger equation. Note that the normalization of the path integral needs to be fixed in exactly the same way as in the free particle case. An arbitrary continuous potential does not affect the normalization, although singular potentials require careful treatment.
Since the states obey the Schrödinger equation, the path integral must reproduce the Heisenberg equations of motion for the averages of x and variables, but it is instructive to see this directly. The direct approach shows that the expectation values calculated from the path integral reproduce the usual ones of quantum mechanics.
Start by considering the path integral with some fixed initial state
Now note that at each separate time is a separate integration variable. So it is legitimate to change variables in the integral by shifting: where is a different shift at each time but , since the endpoints are not integrated:
The change in the integral from the shift is, to first infinitesimal order in epsilon:
which, integrating by parts in t, gives:
But this was just a shift of integration variables, which doesn't change the value of the integral for any choice of . The conclusion is that this first order variation is zero for an arbitrary initial state and at any arbitrary point in time:
If the action contains terms which multiply and , at the same moment in time, the manipulations above are only heuristic, because the multiplication rules for these quantities is just as noncommuting in the path integral as it is in the operator formalism.
The formulation of the path integral does not make it clear at first sight that the quantities x and p do not commute. In the path integral, these are just integration variables and they have no obvious ordering. Feynman discovered that the non-commutativity is still there .
To see this, consider the simplest path integral, the brownian walk. This is not yet quantum mechanics, so in the path-integral the action is not multiplied by i:
The quantity x(t) is fluctuating, and the derivative is defined as the limit of a discrete difference.
Note that the distance that a random walk moves is proportional to , so that:
The quantity is ambiguous, with two possible meanings:
In ordinary calculus, the two are only different by an amount which goes to zero as goes to zero. But in this case, the difference between the two is not zero:
give a name to the value of the difference for any one random walk:
and note that is a rapidly fluctuating statistical quantity, whose average value is 1, i.e. a normalized "Gaussian process". The fluctuations of such a quantity can be described by a statistical Lagrangian , and the equations of motion for f derived from extremizing the action S corresponding to just set it equal to 1. In physics, such a quantity is "equal to 1 as an operator identity". In mathematics, it "weakly converges to 1". In either case, it is 1 in any expectation value, or when averaged over any interval, or for all practical purpose.
Defining the time order to be the operator order:
This is called the Ito lemma in stochastic calculus, and the (euclideanized) canonical commutation relations in physics.
For a general statistical action, a similar argument shows that
For a particle in curved space the kinetic term depends on the position and the above time slicing cannot be applied, this being a manifestation of the notorious operator ordering problem in Schrödinger quantum mechanics. One may, however, solve this problem by transforming the time-sliced flat-space path integral to curved space using a multivalued coordinate transformation (nonholonomic mapping explained here).
The path integral is just the generalization of the integral above to all quantum mechanical problems—
The connection with statistical mechanics follows. Considering only paths which begin and end in the same configuration, perform the Wick rotation , i.e., make time imaginary, and integrate over all possible beginning/ending configurations. The path integral now resembles the partition function of statistical mechanics defined in a canonical ensemble with temperature . Strictly speaking, though, this is the partition function for a statistical field theory.
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
For instance, if
The path integral formulation was very important for the development of quantum field theory. Both the Schrödinger and Heisenberg approaches to quantum mechanics single out time, and are not in the spirit of relativity. For example, the Heisenberg approach requires that scalar field operators obey the commutation relation
for x and y two simultaneous spatial positions, and this is not a relativistically invariant concept. The results of a calculation are covariant at the end of the day, but the symmetry is not apparent in intermediate stages. If naive field theory calculations did not produce infinite answers in the continuum limit, this would not have been such a big problem--- it would just have been a bad choice of coordinates. But the lack of symmetry means that the infinite quantities must be cut off, and the bad coordinates make it nearly impossible to cut off the theory without spoiling the symmetry. This makes it difficult to extract the physical predictions, which require a careful limiting procedure.
The problem of lost symmetry also appears in classical mechanics, where the Hamiltonian formulation also superficially singles out time. The Lagrangian formulation makes the relativistic invariance apparent. In the same way, the path integral is manifestly relativistic. It reproduces the Schrödinger equation as the evolution equation for the state at each time slice, and it also includes the Heisenberg equations of motion as a differential identity obeyed by averages of the integration variables at nearby times. It includes the canonical commutation relations in a natural way, and it extends them using the relativistic symmetry to operator product rules which are new relations difficult to extract from the old formalism.
Further, different choices of canonical variables lead to very different seeming formulations of the same theory. The transformations between the variables can be very complicated, but the path integral makes them into reasonably straightforward changes of integration variables. For these reasons, the Feynman path integral has made earlier formalisms largely obsolete.
The price of a path integral representation is that the unitarity of a theory is no longer self evident, but it can be proven by changing variables to some canonical representation. 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.
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. Superposing different values of the initial position with an arbitrary initial state constructs the final state.
For a spatially homogenous system, where K(x, y) is a only a function of (x-y), the integral is a convolution, the final state is the initial state convolved with the propagator.
For a free particle of mass m, the propagator can be evaluated either explicitly from the path integral or by noting that the Schrödinger equation is a diffusion equation in imaginary time and the solution must be a normalized Gaussian:
Taking the Fourier transform in (x-y) produces another Gaussian:
and in p-space the proportionality factor here is constant in time, as will be verified in a moment. The Fourier transform in time, extending K(p;T) to be zero for negative times, gives the Green's Function, or the frequency space propagator:
Which is the reciprocal of the operator which annihilates the wavefunction in the Schrödinger equation, which wouldn't have come out right if the proportionality factor weren't constant in the p-space representation.
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 on G. The infinitesimal term can be interpreted as an infinitesimal rotation toward imaginary time.
It is also possible to reexpress the nonrelativistic time evolution in terms of propagators which go toward the past, since the Schrödinger equation is time-reversible. The past propagator is the same as the future propagator except for the obvious difference that it vanishes in the future, and in the gaussian is replaced by . 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 which 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 arclength of the path of an oscillating quantity, and like the nonrelativistic path integral should be interpreted as slightly rotated into imaginary time. The function K(x-y,tau) can be evaluated when the sum is over paths in Euclidean space.
This describes a sum over all paths of length of the exponential of minus the length. This can be given a probability interpretation. The sum over all paths is a probability average over a path constructed step by step. The total number of steps is proportional to , and each step is less likely the longer it is. By the central limit theorem, the result of many independent steps is a Gaussian of variance proportional to .
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. Taking a Fourier transform over the variable can be done for each value of separately, and because each separate contribution is a Gaussian, gives whose fourier transform is another Gaussian with reciprocal width. So in p-space, the propagator can be reexpressed simply:
Which is the Euclidian propagator for a scalar particle. Rotating to be imaginary gives the usual relativistic propagator, up to a -i and an ambiguity which will be clarified below.
This expression can be interpreted in the nonrelativistic limit, where it is convenient to split it by partial fractions:
For states where one nonrelativistic particle is present, the initial wavefunction has a frequency distribution concentrated near . 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. For frequencies near , the dominant first term has the form:
The second term has a nonrelativistic limit also, but this limit is concentrated on frequencies which 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.
The proper way to express this mathematically is that, adding a small suppression factor in proper time, the limit where of the first term must vanish, while the limit of the second term must vanish. In the fourier transform, this means shifting the pole in 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 . The terms can be recombined:
Which when factored, produces opposite sign infinitesimal terms in each factor. This is the mathematically precise form of the relativistic particle propagator, free of any ambiguities. The term introduces a small imaginary part to the , which in the Minkowski version is a small exponential suppresion of long paths.
So in the relativistic case, the Feynman path-integral representation of the propagator includes paths which go backwards in time, which describe antiparticles. The paths which 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 which are nonzero outside the lightcone, meaning that it is impossible to keep a particle from travelling faster than light. Such a particle cannot be have a Greens 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. The action is referred to technically as a functional of the field: where the field 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 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.
In quantum field theory, if the action is given by the functional 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
The symbol 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.
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 (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 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
which now becomes
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 for the expectation:
for any polynomially bounded functional F.
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:
Basically, if 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 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
and G is a functional of J, then
If the functional measure is not translationally invariant, it might be possible to express it as the product where M is a functional and 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 in this equation by another functional
where S is the Wick-rotated classical action of the particle, J is the classical action with an extra term "x" and delta here is the functional derivative operator
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 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 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.
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
(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 are not insisting upon the gauge principle), but just that Q is. And we also assume the even stronger assumption that the functional measure is locally invariant:
Then, we would have
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
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 (see the reference below) claim the interpretation explains the Einstein-Podolsky-Rosen paradox without resorting to nonlocality. (Note that the Copenhagen/pragmatism interpretation claims there is no paradox--only a sloppy materialism motivated question on the part of EPR--Joseph Wiener a lecture. On the other hand, the fact that the EPR thought experiment (and its result) does represent the results of a QM experiment says that (despite the path dependence of parallelness/anti-parallelness in curved space) all contributions of paths close to black holes cancel in the action for an EPR style experiment here on earth.)
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.