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I was reading the book 'Gauge Fields and Strings' by A. Polyakov and I don't understand a step in the 1D single particle propagator derivation..

The part I don't understand is Eq. (1.2). for the transition amplitude/propagator:

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Where do the coefficients $\left ( \dfrac{m}{2 \pi \text i \hbar (t_j - t_{j-1})} \right )^{1/2}$ and $ \left ( \dfrac{m}{2 \pi \text i \hbar (T - t_{N-1})} \right )^{1/2}$ come from? I haven't taken QFT yet so I'm not familiar with path integrals. I've studied the derivation for Eq. (1.1) but don't understand the steps to Eq. (1.2). I'm grateful for any help or suggestions.

Qmechanic
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Saarnamoinen
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1 Answers1

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The square root factors in eq. (1.2) are the celebrated Feynman fudge factors. They can be understood in the Hamiltonian phase space path integral as arising from a Gaussian momentum integration, cf. e.g. my Phys.SE answer here.

References:

  1. R.P. Feynman & A.R. Hibbs, Quantum Mechanics and Path Integrals, 1965; chapters 2 + 3 + 4.

  2. W. Dittrich & M. Reuter, Classical and Quantum Dynamics, 6th ed, 2020; chapters 19 + 20.

Qmechanic
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