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## ISBN 13: 9780750308014

Limit the search to the library catalogue. Direct access to the library catalogue. Chaichian and A. Path Integrals in Physics: Volume I, Stochastic Processes and Quantum Mechanics presents the fundamentals of path integrals, both the Wiener and Feynman type, and their many applications in physics.

Accessible to a broad community of theoretical physicists, the book deals with systems possessing a infinite number of degrees in freedom.

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It discusses the general physical background and concepts of the path integral approach used, followed by a detailed presentation of the most typical and important applications as well as problems with either their solutions or hints how to solve them. It describes in detail various applications, including systems with Grassmann variables. 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.

In quantum mechanics, as in classical mechanics, the Hamiltonian is the generator of time translations. For states with a definite energy, this is a statement of the de Broglie relation between frequency and energy, and the general relation is consistent with that plus the superposition principle. The Hamiltonian in classical mechanics is derived from a Lagrangian , which is a more fundamental quantity relative to special relativity. The Hamiltonian indicates how to march forward in time, but the time is different in different reference frames. The Lagrangian is a Lorentz scalar , while the Hamiltonian is the time component of a four-vector.

So the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics. The Hamiltonian is a function of the position and momentum at one time, and it determines the position and momentum a little later. The Lagrangian is a function of the position now and the position a little later or, equivalently for infinitesimal time separations, it is a function of the position and velocity.

The relation between the two is by a Legendre transformation , and the condition that determines the classical equations of motion the Euler—Lagrange equations is that the action has an extremum. In quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory.

## Path Integrals in Physics - Masud Chaichian, A Demichev - Bok () | Bokus

In classical mechanics, with discretization in time, the Legendre transform becomes. The inverse Legendre transform is. In quantum mechanics, the state is a superposition of different states with different values of q , or different values of p , and the quantities p and q can be interpreted as noncommuting operators. The operator p is only definite on states that are indefinite with respect to q.

So consider two states separated in time and act with the operator corresponding to the Lagrangian:. If the multiplications implicit in this formula are reinterpreted as matrix multiplications, the first factor is.

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That is the action on the Hilbert space — change basis to p at time t. This is not very different from just ordinary time evolution: the H factor contains all the dynamical information — it pushes the state forward in time. The first part and the last part are just Fourier transforms to change to a pure q basis from an intermediate p basis. Another way of saying this is that since the Hamiltonian is naturally a function of p and q , exponentiating this quantity and changing basis from p to q at each step allows the matrix element of H to be expressed as a simple function along each path.

This function is the quantum analog of the classical action. This observation is due to Paul Dirac. Dirac further noted that one could square the time-evolution operator in the S representation:. While in the H representation the quantity that is being summed over the intermediate states is an obscure matrix element, in the S representation it is reinterpreted as a quantity associated to the path.

In the limit that one takes a large power of this operator, one reconstructs the full quantum evolution between two states, the early one with a fixed value of q 0 and the later one with a fixed value of q t. The result is a sum over paths with a phase, which is the quantum action. Crucially, Dirac identified in this article the deep quantum-mechanical reason for the principle of least action controlling the classical limit see quotation box.

This was done by Feynman.

Feynman showed that Dirac's quantum action was, for most cases of interest, simply equal to the classical action, appropriately discretized. This means that the classical action is the phase acquired by quantum evolution between two fixed endpoints. He proposed to recover all of quantum mechanics from the following postulates:. In order to find the overall probability amplitude for a given process, then, one adds up, or integrates , the amplitude of the 3rd postulate over the space of all possible paths of the system in between the initial and final states, including those that are absurd by classical standards.

## Path Integrals and Quantum Anomalies

In calculating the probability amplitude for a single particle to go from one space-time coordinate to another, it is correct to include paths in which the particle describes elaborate curlicues , curves in which the particle shoots off into outer space and flies back again, and so forth. The path integral assigns to all these amplitudes equal weight but varying phase , or argument of the complex number. Contributions from paths wildly different from the classical trajectory may be suppressed by interference see below. Feynman showed that this formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics when the Hamiltonian is at most quadratic in the momentum.

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.

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A Feynman diagram is a graphical representation of a perturbative contribution to the transition amplitude. One common approach to deriving the path integral formula is to divide the time interval into small pieces. Once this is done, the Trotter product formula tells us that the noncommutativity of the kinetic and potential energy operators can be ignored. For a particle in a smooth potential, the path integral is approximated by zigzag paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position x a at time t a to x b at time t b , the time sequence.

Actually L is the classical Lagrangian of the one-dimensional system considered,. Thus, in contrast to classical mechanics, not only does the stationary path contribute, but actually all virtual paths between the initial and the final point also contribute. In terms of the wave function in the position representation, the path integral formula reads as follows:.

The path integral representation gives the quantum amplitude to go from point x to point y as an integral over all paths. 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 amplitude or Kernel reads:. The result is easy to evaluate by taking the Fourier transform of both sides, so that the convolutions become multiplications:. The Fourier transform of the Gaussian G is another Gaussian of reciprocal variance:. 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.

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The probability interpretation gives a natural normalization choice. The path integral should be defined so that. For oscillatory path integrals, ones with an i in the numerator, the time slicing produces convolved Gaussians, just as before.