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Kevin_Axion
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In a Feynman Path integral, [tex]Z(\phi) = \int \cal D \phi[/tex] [tex] e^{\frac{iS(x)}{\hbar}}[/tex], what does the object [tex]e^{\frac{iS(x)}{\hbar}}[/tex] mean?
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Kevin_Axion said:Is it correct to say that [tex]e^{\frac{iS[\phi]}{\hbar}}[/tex] is similar to the traditional formula [tex]e^{ix} = cosx + isinx[/tex] except now x is a functional (action) and there is a factor of [tex]\frac{1}{\hbar}[/tex] do to quantization?
A. Neumaier said:The integral of e^{ix^2} over R is neither Riemann integrable nor Lebesgue integrable,
and Henstock integrability, which fits here, is not common knowledge. In infinite dimensions, the difficulties are much worse...
Therefore one often (e..g., in lattice simulations of QFT) uses so-called Euclidean path integrals, where i is replaced by -1 to make it mathematically tractable. This can be justified to some extent by analytic continuation.
RedX said:When you replace i by -1, paths with high action no longer vary wildly in phase and hence cancel, but instead become suppressed.
This to me is radically different. In the former case, there is a collective canceling of high action paths. In the latter case, there is individual suppression of each individual high action path.
Is this the right interpretation of the difference between working in Minkowski space and Euclidean space?
A. Neumaier said:Yes. This means that the latter is well-behaved and can be studied by much more powerful techniques than the former, where one mainly relies on purely formal manipulations that are known to be sometimes wrong in much simpler situations.
if you replace the i by a variable j then you can interpolate between the regimes. For j=-1 one has the Euclidean integral and as one shifts more and more of the real part to the imaginary part one tends to the Minkowski integral. In the limit j-> i one gets the latter.
This is called analytic continuation, and in all cases where it can be proved to be valid, things are fine. The precise conditions are determined by the Osterwalder-Schrader theorem.
RedX said:It seems to me that [tex]e^{\int -\frac{\mathcal H}{\hbar}d\tau}[/tex] is much easier to understand than [tex]e^{\int i\frac{\mathcal L}{\hbar}dt}=e^{iS} [/tex] when you take the limit [tex] \hbar[/tex] goes to zero (i.e., when the action is really big compared to [tex]\hbar [/tex] ), yet most popular books insist on the latter and talk about phase cancellations at large actions (Minkowski) instead of supression because of too much energy on the path (Euclidean).
RedX said:Actually if you take the Hamiltonian H in the path integral and replace it by [tex]H(1-i\epsilon)[/tex] for infinitismally small positive [tex]\epsilon [/tex], then you get the [tex]i\epsilon [/tex] prescription which can take the place of a Wick rotation. Is this the same thing as your variable complex variable j?
The Feynman Path Integral is a mathematical framework developed by physicist Richard Feynman to describe the behavior of particles in quantum mechanics. It is based on the principle of least action, where the path a particle takes is determined by minimizing the action (a mathematical quantity related to the energy) of the system.
The Feynman Path Integral can be used to calculate the probability amplitude of a particle moving from one point to another in spacetime. This probability amplitude is represented by
In quantum mechanics,
Yes, the Feynman Path Integral can be applied to all physical systems, as long as they can be described by the principles of quantum mechanics. This includes subatomic particles, atoms, and even large-scale systems like black holes. However, the calculations can become increasingly complex for more complex systems.
One limitation of the Feynman Path Integral is that it does not take into account the effects of gravity, which is described by the theory of general relativity. This has led to attempts to combine the Feynman Path Integral with general relativity to create a theory of quantum gravity. Additionally, some critics argue that the Feynman Path Integral is not a fully rigorous mathematical framework and may lead to ambiguous or incorrect results in certain cases.