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I am studying textbook by Schulman od path integration. In first chapters of this book he uses Lagrangian form of the path integral and arrives at the conclusion that when a particle is moving in external gauge field one needs to use midpoint rule for evaluation of $\vec A (\vec x)$ in path integrals. I'd like to compare that to what is done in Weinberg (quantum fields). He explains there that choice of the evaluation point is equivalent to choosing ordering of operators in quantum counterpart of classical function. I followed steps made there and found that we can construct path integrals evaluated using the endpoint rule provided that we take classical Hamiltonian as $$ H_{eff}= \frac{(\vec p - q \vec A)^2 + iq \vec \nabla A}{2m} + q V (\vec x). $$ If we perform Gaussian integration in Hamiltonian version of path integral we get Lagrangian formulation with Lagrangian $$ L_{eff} = \frac{m}{2} \dot{\vec{x}}^2 + q \vec A \cdot \dot{\vec x} - \frac{iq}{2m} \vec{\nabla} \cdot \vec A - q V (\vec x) .$$ It can be verfied straightforwardly that with this choice of Lagrangian usual path integral formula approximated with saddle point method produces correct results in the sense that the wave-function satisfies Shrodinger equation with correct Hamiltonian operator. At first it seems that everything is in order, but actually not quite! The problem is that the action obtained with these $L$ or $H$ does not transform "as it should" under gauge transformations. It seems that something important is lost here.

Question: Can one really treat explanations in Weinberg seriously? What is the exact relation between evaluation point in the path integral and ordering ambiguities in quantization? Is there a way to preserve gauge covariance obtained by the procedure presented in Weinberg (with endpoint rule)?

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  • $\begingroup$ Which pages in Schulman & Weinberg? $\endgroup$ – Qmechanic Nov 11 '16 at 20:32
  • $\begingroup$ Schulman section four "Vector Potentials and Another Proof of the Path Integral Formula" discusses path integral for particle in magnetic field. In my version of the book it's page 22. In Weinberg vol. 1 in first section of chapter nine "Functional integral methods" (this is my translation of the title from polish to english) he discusses general structure of path integral quantization (but not this specific example I discuss here). It is difficult for me to renoncile these two approaches. $\endgroup$ – Blazej Nov 12 '16 at 11:40
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Weinberg [1] does only discuss the naive formal phase space path integral, which ignores operator ordering problems. For a more careful analysis see e.g. Schulman [2] and links in this Phys.SE post.

Concerning the Hamiltonian formulation of a non-relativistic charged particle in an E&M background field, note that the canonical momentum $\hat{\bf p}$ transforms under gauge transformations to restore gauge covariance, cf. e.g. Weinberg [3].

References:

  1. S. Weinberg, Quantum Theory of Fields, Vol. 1, 1995; Section 9.1.

  2. L.S. Schulman, Techniques and applications of path integration, 1981; Chap. 4 & 5.

  3. S. Weinberg, Lectures on Quantum Mechanic, 2012; Section 10.2.

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  • $\begingroup$ I know about transformation of $p$ under gauge transformations. However actually I worked with the Lagrangian itself. Under gauge transformations it should transform to itself plus a total derivative, which is not the case for $L$ I wrote down due to the term with $\vec{\nabla} \cdot \vec A$. Is it correct to conclude that in this case approach presented by Weinberg is just wrong? $\endgroup$ – Blazej Nov 12 '16 at 16:20
  • $\begingroup$ Let me put it this way: you are reading a textbook beyond its intended introductory scope. Or phrased alternatively: any physics book comes with an implicit disclaimer that it is up to its reader to understand its limitations. $\endgroup$ – Qmechanic Nov 12 '16 at 17:03

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