Path integrals Definition and 60 Threads

The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.
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. Another advantage is that it is in practice easier to guess the correct form of the Lagrangian of a theory, which naturally enters the path integrals (for interactions of a certain type, these are coordinate space or Feynman path integrals), than the Hamiltonian. Possible downsides of the approach include that unitarity (this is related to conservation of probability; the probabilities of all physically possible outcomes must add up to one) of the S-matrix is obscure in the formulation. The path-integral approach has been proved to be equivalent to the other formalisms of quantum mechanics and quantum field theory. Thus, by deriving either approach from the other, problems associated with one or the other approach (as exemplified by Lorentz covariance or unitarity) go away.The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s, which unified quantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition. The Schrödinger equation is a diffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possible random walks.The basic idea of the path integral formulation can be traced back to Norbert Wiener, who introduced the Wiener integral for solving problems in diffusion and Brownian motion. This idea was extended to the use of the Lagrangian in quantum mechanics by Paul Dirac in his 1933 article. The complete method was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier in his doctoral work under the supervision of John Archibald Wheeler. 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.

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  1. R

    Weinberg, Vol. 1: Path Integrals Query

    I am trying to study QFT from Weinberg's Vol. 1. I am at the moment stuck at the path integral quantization of QED (Weinberg's treatment). I am not sure how he integrates out the matter field momenta (for the spinor field) in eq. (9.6.5). I thought for spinor fields you don't do that...
  2. K

    Path integrals and propagators

    we know that for the SE equation we find the propagator (i\hbar \partial _{t} - \hbar ^{2} \nabla +V(x,y,z) )K(x,x')=\delta (x-x') with m=1/2 for simplicity then we know that the propagator K(x,x') may be obtained from the evaluation of the Path integral. K(x,x')=C \int \mathcal...
  3. B

    Path Integrals and Non-Abelian Gauge Theories

    Ok, I have a question about this Fade'ev Popov procedure of teasing out the ghosts when one quantizes a non-Abelian gauge theory with path integrals. The factor of 1 that people insert, for some gauge fixing function f, and some non-Abelian symmetry \mathcal{G} is: 1=\int \mathcal{D}U...
  4. S

    Mathematica Mathematical Physics. Path integrals

    Hello all I need some special help concerning the path integrals and exactely about the techniques of Fradkin-Gitman and also the technique of Alexandrou et al., what's they're exactely about ?. (what does it mean here al. in "Alexandrou et al." ):smile: Thank you very much for every...
  5. P

    Why Should You Study Feynman Path Integrals in Quantum Mechanics?

    My QM prof skipped over the topic of the Feynman Path integral formulation... Is this material important enough that I should learn it on my own (personal curiosity aside)? The Text is Principles of Quantum Mechanics by R. Shankar
  6. K

    Perturbation theory and Path integrals.

    Let's suppose we have a theory with Lagrangian: \mathcal L_{0} + gV(\phi) where the L0 is a quadratic Lagrangian in the fields then we could calculate 'exactly' the functional integral: \int\mathcal D[ \phi ]exp(iS_{0}[\phi]/\hbar+gV(\phi)) where J(x) is a source then we could...
  7. R

    What does a path integral measure in complex analysis?

    What exactly does a path integral measure? Is it area between the ends/bounds of the line? Or is it the length of the line? Just started complex analysis and am comletely confused by this.
  8. C

    Proving a force is conservative (via path integrals)

    It's always the easy questions that get me stuck... For some reason, I'm having a mental block on how to answer this one: Consider the force function: F = ix + jy Verify that it is conservative by showing that the integral, \int F \cdot dr is independent of the path of integration by...
  9. M

    Path integrals and the sum of surfaces, is this general?

    I'm wondering if it is true that any surface can be equated to a weighted sum of a basis of surfaces differring only by genus? I think this is asking whether the path integral formulation for strings is more general. Thanks.
  10. E

    Solve Path Integrals: Find Schroedinger Equation & Quantum Gravity

    If you have path integrals could you find the schroedinguer equation?..in fact why is not this made to find the equation in quantum gravity?..
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