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26

In order to use Lagrangians in QM, one has to use the path integral formalism. This is usually not covered in a undergrad QM course and therefore only Hamiltonians are used. In current research, Lagrangians are used a lot in non-relativistic QM. In relativistic QM, one uses both Hamiltonians and Lagrangians. The reason Lagrangians are more popular is that ...


22

The fundamental equation which serves as the basis for the path-integral formulation of finance and many physical problems is the Chapman-Kolmogorov equation. $$p(X_f|X_i)=\int p(X_f|X_k)p(X_k|X_i) dX_k $$ This is analogous to the following equation for amplitudes in quantum mechanics $$\langle X_f|X_i \rangle=\int \langle X_f|X_k\rangle\langle ...


17

Path integral is indeed very problematic on its own. But there are ways to almost capturing it rigorously. Wiener process One way is to start with Abstract Wiener space that can be built out of the Hamiltonian and carries a canonical Wiener measure. This is the usual measure describing properties of the random walk. Now to arrive at path integral one has ...


17

This is just a property of Gaussian averaging analogous to the finite dimensional case: $\langle e^{ix} \rangle=\frac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^{\infty} e^{ix}e^{-\frac{x^2}{2\sigma^2}}=e^{-\frac{\sigma^2}{2}}= e^{-\frac{\langle x^2 \rangle}{2}}$ The field can be decomposed into its independent Gaussian modes and integrated for each mode ...


13

In 2-dimensional space-time, Feynman path integrals are perfectly well-defined, though understanding how this is done rigorously is somewhat heavy-going. But everything is spelled out in the book ''Quantum Physics: A Functional Integral Point of View'' by Glimm and Jaffe. http://www.amazon.com/Quantum-Physics-Functional-Integral-Point/dp/0387964770 In 4 ...


12

There are already several good answers. Here I will only answer the very last question, i.e., if the Boltzmann factor in the path integral is $f(S(t_f,t_i))$, with action $S(t_f,t_i)=\int_{t_i}^{t_f} dt \ L(t)$, why is the function $f:\mathbb{R}\to\mathbb{C}$ an exponential function, and not something else? Well, since the Feynman "sum over histories" ...


12

There are rigorous constructions of QFTs in infinite volume. Glimm & Jaffe's book does this for interacting 2d scalars (with the assumption that the interactions are not too strong). I'm sure you can find other examples in the literature (or perhaps someone else will point you to them). However, restricting yourself to a lattice doesn't really buy ...


11

Mathematically, a path integral is a generalization of a multi-dimensional integral. In usual $N$-dimensional integrals, one integrates $$\int dx_1 dx_2 \dots dx_N $$ over a subspace of ${\mathbb R}^N$, an $N$-dimensional integral. A path integral is an infinite-dimensional integral $$ \int {\mathcal D}f(y)\, Z[f(y)] $$ over all possible functions $f(y)$ of ...


10

Good question; I remember spending hours trying to understand this when I first learned QFT. Let's address your two main points in turn. First, you say I don't understand how rhyme these two different pictures. Let's outline how to connect the two pictures in steps. It's a good exercise to try and work through all of the gory details yourself, so I ...


10

Comments to the question (v2): 1) The correspondence between Lagrangian (L) and Hamiltonian (H) theories is mired with subtleties. Some general tools for singular Legendre transformations are available, such as Dirac-Bergmann analysis, Faddeev-Jackiw method, etc. But rather than claiming complete understanding and existence of the L-H correspondence, it is ...


10

If the functional derivative $$\tag{1} \frac{\delta F[\phi]}{\delta\phi^{\alpha}(x)} $$ exists (wrt. to a certain choice of boundary conditions), it obeys infinitesimally $$\tag{2}\delta F ~:=~ F[\phi+\delta\phi]- F[\phi] ~=~\int_M \!dx\sum_{\alpha\in J} \frac{\delta F[\phi]}{\delta\phi^{\alpha}(x)}\delta\phi^{\alpha}(x). $$ OP's functional integral ...


10

The book Quantum dissipative systems by Weiss dedicates a subsection to the Feynman Vernon method, see also the original reference. See also this article and chapter 18.8 of the book by Kleinert. It's applied to the Caldeira-Leggett model, which is a toy model for a particle in contact with a heat bath. There are a number of mesoscopic systems out there in ...


9

Yes, for this particular Matsubara sum you mentioned, taking different limits will lead to two results differed by 1. This is because the summation in consideration does not converge, which can be seen from the following integral (the continuous limit of the sum) considering the ultraviolet divergence (the large $\omega$ behavior) ...


9

The easiest way to see imaginary time used is in elementary quantum mechanics in one dimension. (This is the explanation cribbed from wikipedia). Suppose we're looking at a tunneling-through-a-barrier problem. We start with the Schrodinger equation: $$ -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+V(x)\psi(x) = E\psi(x) $$ Make the ansatz $$ \psi(x) ...


9

I think it will depend the kind of statistical mechanics. For classical statistical mechanics, there is no time, so it is really hard to imagine a nice physical picture of the propagation of something. But nevertheless we still talk of loops as propagating "particles" (we give the "momenta", for instance, which is conserved, etc.). Interestingly, ...


9

In the context of axiomatic quantum field theory, there is a theorem (see theorem 3-7 in PCT, Spin and Statistics, and All That by Streater and Wightman, who I will refer to as "SW"), which SW call the "reconstruction theorem," essentially stating that correlation functions serve to completely determine a corresponding field theory in the Hilbert Space ...


8

I) OP is right, ideologically speaking. Ideologically, OP's first eq. $$ \tag{1} \left| \int_{\mathbb{R}}\! \mathrm{d}x_f~K(x_f,t_f;x_i,t_i) \right| ~\stackrel{?}{=}~1 \qquad(\leftarrow\text{Turns out to be ultimately wrong!}) $$ is the statement that a particle that is initially localized at a spacetime event $(x_i,t_i)$ must with probability 100% be ...


8

One of the avenues to search for an answer is the so-called Keldysh formalism which is used extensively in condensed matter, in particular in mescopic physics, to define and study steady-state and time-dependent quantum phenomena in systems with infinitely many degrees of freedom. A recent comprehensive review is given by Kamenev and Levchenko, ...


8

As Weinberg points in his QFT book, in the Hamiltonian formalism it is easier to check the unitarity of the theory because unitarity is directly related to evolution, while in the Lagrangian formalism the symmetries that mix space with time are more explicit. Therefore the Hamiltonian formalism is usually more convenient in non-relativistic and galilean ...


7

Let's do what Heidar says and write it with indices, and identify the Lagrangian. $$ L=\frac{1}{2}(\vec{\nabla}\times \vec{A})^2 = \frac{1}{2}\epsilon_{ijk}\partial_j A_k \epsilon_{ilm}\partial_l A_m $$ where, if you haven't heard of it yet, you pretend there is a summation symbol for each repeated index. Then since there are no bare $A_i$ sitting by ...


7

Nope, Feynman's path integral formulation of quantum mechanics is a method to directly calculate the complex probability amplitudes and all objects that appear in its formalism - not counting proofs of equivalence with other approaches to quantum mechanics - are $c$-numbers representing classical observables. In particular, the exponent in the path integral ...


7

I will add to twistor59 answer. Hawking liked the concept of imaginary time $\tau=\mathrm{i}t$ because it transforms a Lorentzian metric $$ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$$ into a four dimensional like Euclidean metric $$ds^2 = +c^2 d\tau^2 + dx^2 + dy^2 + dz^2$$ Hawking and others believed that a quantum gravity theory could be developed in this ...


7

One major difficulty with defining path-integrals (which is entirely mathematicians' fault)is that the mathematicians insist for no good reason (and many bad ones) that there are non-measurable subsets of R. This is a psychological artifact of early days of set theory, where ZFC ws not seen as a way of generating countable models of a set-theoretic universe, ...


7

Our plan of an answer is as follows. Firstly, we will introduce Planck's constant $\hbar$ so that the particular value $\hbar=1$ corresponds to the original problem. Secondly, we mention a connection to (what physicists often calls) the group property of Feynman path integrals. Thirdly, we will show that the sought-for formula happens to be the classical ...


7

You start by writing down the probability to find a particle at $y$ at time $t$ when it was at $x$ at time $0$, denoted as $K(y,t;x,0)$. You get this by solving the Schrödinger equation with the initial condition $\psi(y,0) = \delta(y-x)$. Then, $K(y,t;x,0) = \psi(y,t)$. Thus, to solve this, we need to know the time development of the initial condition ...


7

The path integral, mathematically speaking, does not exist as an integral: It is not associated with any positive or complex measure. Conversely, the Euclidean path integral does exist. The Wick rotation is a way to "construct" the Feynman integral as a limit case of the well-defined Euclidean one. If, instead, you are interested in an axiomatic approach ...


7

Yes, one traditional alternative to the path integral formalism is the operator formalism. For QED with abelian gauge group, the old quantization formulation is the Gupta-Bleuler formulation. For QCD/Yang-Mills theory with non-abelian gauge group, the Gupta-Bleuler formulation is replaced by the BRST formulation. The BRST formulation exists in at least 3 ...


6

Why do we gauge-fix the path integral in the first place? If we were doing lattice gauge theory, we didn't need to gauge-fix. But in the continuum case, (the Hessian of) the action for a generalized$^1$ gauge theory has zero-directions that lead to infinite factors when performing the path integral over gauge orbits. In a BRST formulation (such as, e.g., the ...


6

1st comment: It's worth thinking for a second about where Wick rotation comes from. You can do this in the context of the quantum mechanics of a free particle. In QFT, all of the details are more complicated, but the basic idea is the same. In free particle, QM, we get the path integral by inserting sums over intermediate states at various times. The ...



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