# Tag Info

29

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 ...

27

Your statement working with and subtracting infinities ... which in general have no mathematical meaning is not really correct, and seems to have a common misunderstanding in it. The technical difficulties from QFT do not come from infinities. In fact, ideas basically equivalent to renormalization and regularization have been used since the ...

24

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 $$\langle X_f|X_i \rangle=\int \langle X_f|X_k\rangle\langle ... 20 Path integral is indeed very problematic on its own. But there are ways to almost capturing it rigorously. Wiener process One way is to start with Abstract Wiener space that can be built out of the Hamiltonian and carries a canonical Wiener measure. This is the usual measure describing properties of the random walk. Now to arrive at path integral one has ... 17 There are already several good answers. Here I will only answer the very last question, i.e., if the Boltzmann factor in the path integral is f(S(t_f,t_i)), with action S(t_f,t_i)=\int_{t_i}^{t_f} dt \ L(t), why is the function f:\mathbb{R}\to\mathbb{C} an exponential function, and not something else? Well, since the Feynman "sum over histories" ... 17 This is just a property of Gaussian averaging analogous to the finite dimensional case: \langle e^{ix} \rangle=\frac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^{\infty} e^{ix}e^{-\frac{x^2}{2\sigma^2}}=e^{-\frac{\sigma^2}{2}}= e^{-\frac{\langle x^2 \rangle}{2}} The field can be decomposed into its independent Gaussian modes and integrated for each mode ... 15 First: There is no rigorous construction of the standard model, rigorous in the sense of mathematics (and no, there is not much ambivalence about the meaning of rigor in mathematics). That's a lot of references that Daniel cited, I'll try to classify them a little bit :-) Axiomatic (synonymous: local or algebraic) QFT tries to formulate axioms for the ... 15 The easiest way to see imaginary time used is in elementary quantum mechanics in one dimension. (This is the explanation cribbed from wikipedia). Suppose we're looking at a tunneling-through-a-barrier problem. We start with the Schrodinger equation:$$ -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+V(x)\psi(x) = E\psi(x) $$Make the ansatz$$ \psi(x) ...

15

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 ...

14

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 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 ...

12

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 ...

12

I actually don't think that this view of light being in a quantum superposition is anything new: what Discover magazine is describing (I believe) is the stock standard picture of how one would describe a system of cells, molecules, chloroplasts, fluorophores, whatever interacting with the quantised electomagnetic field. My simplified account here (answer to ...

11

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, ...

11

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) ...

11

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 ...

11

This type of problems is often referred to as constrained mechanical system. It was studied by Dirac, who developed the theory of constrained quantization. This theory was formalized and further developed by Marseden and Weinstein to what is called "Symplectic reduction". A particularly illiminating chapter for finite dimensional systems may be found in ...

11

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

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

For a given quantum system, the kernel of the path integral is, in fact, the kernel of an integral transform as you explicitly write down. It is the transform that governs time evolution of the system as is manifest in your first equation. For this reason, it is often referred to as the propagator of a given system. For example, for a single, ...

9

The role of coherence in biological electron transport, e.g. within chromophores, is an open and actively researched problem in quantum optics/quantum chemistry. The two classic theoretical treatments which kick-started the field are by Plenio & Huelga and Mohseni et al.. Since then an enormous literature has emerged on the topic. A basic, generic model ...

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 ...

9

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 ...

9

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 ...

9

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 ...

9

Here is my answer from a condensed matter physics point of view: Quantum field theory is a theory that describes the critical point and the neighbor of the critical point of a lattice model. (Lattice models do have a rigorous definition). So to rigorously define/classify quantum field theories is to classify all the possible critical points of lattice ...

9

The theory of deformation quantization provides a framework in which the quantum to classical transition can be carried out and understood. According to this theory, for (practically any) quantum system, one can find (may be nonuniquely) a Poisson manifold $\mathcal{M}$ (phase space) equipped with an associative product called the "star product" such that ...

8

A way to understand this, is to imagine that light actually follows all paths. However, most paths experience destructive interference with other paths. The only paths that do not experience destructive interference are those in the neighbourhood of paths with stationary (e.g., minimal) action (time). I strongly recommend reading Feynman's QED: The Strange ...

8

The path integral over a "thick layer" of spacetime always produces the transition amplitudes $$\langle {\rm final}| U | {\rm initial}\rangle$$ where $U$ is the appropriate unitary evolution operator. This is already true in non-relativistic quantum mechanics where the equivalence between Feynman's path integral approach and the operator formalism is being ...

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