Path integral formulation

This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of canonical quantization.

Unlike previous methods, the path integral allows one to easily change coordinates between very different canonical descriptions of the same quantum system.

[6] This idea was extended to the use of the Lagrangian in quantum mechanics by Paul Dirac, whose 1933 paper gave birth to path integral formulation.

The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the Wheeler–Feynman absorber theory using a Lagrangian (rather than a Hamiltonian) as a starting point.

Dirac's work did not provide a precise prescription to calculate the sum over paths, and he did not show that one could recover the Schrödinger equation or the canonical commutation relations from this rule.

The path integral formulation of quantum field theory represents the transition amplitude (corresponding to the classical correlation function) as a weighted sum of all possible histories of the system from the initial to the final state.

[nb 1] In the limit n → ∞, this becomes a functional integral, which, apart from a nonessential factor, is directly the product of the probability amplitudes ⟨xb, tb|xa, ta⟩ (more precisely, since one must work with a continuous spectrum, the respective densities) to find the quantum mechanical particle at ta in the initial state xa and at tb in the final state xb.

To do this, it is convenient to start without the factor i in the exponential, so that large deviations are suppressed by small numbers, not by cancelling oscillatory contributions.

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.

Only after replacing the time t by another path-dependent pseudo-time parameter the singularity is removed and a time-sliced approximation exists, which is exactly integrable, since it can be made harmonic by a simple coordinate transformation, as discovered in 1979 by İsmail Hakkı Duru and Hagen Kleinert.

Since the time separation is infinitesimal and the cancelling oscillations become severe for large values of ẋ, the path integral has most weight for y close to x.

The direct approach shows that the expectation values calculated from the path integral reproduce the usual ones of quantum mechanics.

If the action contains terms that multiply ẋ and x, 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.

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.

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 L just set it equal to 1.

For a general statistical action, a similar argument shows that and in quantum mechanics, the extra imaginary unit in the action converts this to the canonical commutation relation, 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.

[17] Much of the study of quantum field theories from the path-integral perspective, in both the mathematics and physics literatures, is done in the Euclidean setting, that is, after a Wick rotation.

The Wick-rotated path integral—described in the previous subsection, with the ordinary action replaced by its "Euclidean" counterpart—now resembles the partition function of statistical mechanics defined in a canonical ensemble with inverse temperature proportional to imaginary time, ⁠1/T⁠ = ⁠ikBt/ħ⁠.

For example, the Heisenberg approach requires that scalar field operators obey the commutation relation for two simultaneous spatial positions x and y, and this is not a relativistically invariant concept.

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.

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.

It is also possible to reexpress the nonrelativistic time evolution in terms of propagators going toward the past, since the Schrödinger equation is time-reversible.

By the central limit theorem, the result of many independent steps is a Gaussian of variance proportional to Τ: 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: where W(Τ) is a weight factor, the relative importance of paths of different proper time.

By the translation symmetry in proper time, this weight can only be an exponential factor and can be absorbed into the constant α: This is the Schwinger representation.

So in the relativistic case, the Feynman path-integral representation of the propagator includes paths going backwards in time, which describe antiparticles.

Unlike the nonrelativistic case, it is impossible to produce a relativistic theory of local particle propagation without including antiparticles.

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.

The sum-over-histories method gives identical results to canonical quantum mechanics, and Sinha and Sorkin[22] claim the interpretation explains the Einstein–Podolsky–Rosen paradox without resorting to nonlocality.

Quantum tunnelling can be modeled by using the path integral formation to determine the action of the trajectory through a potential barrier.

Using the WKB approximation, the tunneling rate (Γ) can be determined to be of the form with the effective action Seff and pre-exponential factor Ao.