Linked Questions

37
votes
2answers
8k views

Equivalence of canonical quantization and path integral quantization

Consider the real scalar field $\phi(x,t)$ on 1+1 dimensional space-time with some action, for instance $$ S[\phi] = \frac{1}{4\pi\nu} \int dx\,dt\, (v(\partial_x \phi)^2 - \partial_x\phi\partial_t \...
22
votes
2answers
3k views

Quantization of a particle on a spherical surface

Suppose we have a particle of mass $m$ confined to the surface of a sphere of radius $R$. The classical Lagrangian of the system is $$L = \frac{1}{2}mR^2 \dot{\theta}^2 + \frac{1}{2}m R^2 \sin^2 \...
19
votes
2answers
7k views

Time ordering and time derivative in path integral formalism and operator formalism

In operator formalism, for example a 2-point time-ordered Green's function is defined as $\langle\mathcal{T}\phi(x_1)\phi(x_2)\rangle_{op}=\theta(x_1-x_2)\phi(x_1)\phi(x_2)+\theta(x_2-x_1)\phi(x_2)\...
15
votes
1answer
4k views

Why path integral approach may suffer from operator ordering problem?

In Assa Auerbach's book (Ref. 1), he gave an argument saying that in the normal process of path integral, we lose information about ordering of operators by ignoring the discontinuous path. What did ...
8
votes
3answers
980 views

Why there is no unique “recipe” for quantization of a classical theory?

I have seen in Wikipedia that different quantization methods exist (see Wiki article with name "Quantization"). Moreover, Wikipedia stated that there is more than one way to quantize a classical ...
16
votes
1answer
2k views

A curious issue about Dyson-Schwinger equation(DSE): why does it work so well?

This question comes out of my other question "Time ordering and time derivative in path integral formalism and operator formalism", especially from the discussion with drake. The original post is ...
13
votes
1answer
1k views

What's the relation between path integral and Dyson series?

If one solves the Schrodinger equation $$i\hbar\partial_tU(t,0) = H U(t,0)$$ for time evolution operator $U(t,0)$, one can get the following Dyson series $$U(t,0) = \sum_n(\dfrac{-i}{\hbar})^n\...
6
votes
2answers
415 views

Path integral and Out-of-time-ordered (OTOC) correlator

A simple observation that any insertions within the path integral are classical variables (Not operators) and hence, objects inside the path integral "commute" (is symmetric under exchange). Hence, ...
2
votes
2answers
272 views

Uniqueness in the path integral vs canonical quantisation

In quantum mechanics it is well known that if you have a Lagrangian $\mathcal{L}$ and you want to quantise it, there is no unique way of doing this. This is because when you construct the Hamiltonian $...
5
votes
2answers
87 views

How do we know which canonical theory the path integral is equivalent to?

So if we have a (classical) theory with fields $\phi$ and conjugate momenta $\phi$, we turn this into a (canonically quantised) quantum theory by promoting these to operators and impose some sort of ...
1
vote
0answers
176 views

Path integrals vs operator

I have a statement that the path integrals formalism is eqivalent to operator formalism in quantum mechanics. Is it a correct statement? I understand that each of these two formalisms has its ...
1
vote
1answer
85 views

Kinetic momentum tensor and causal propagator

I can't find anywhere how the kinetic operator defined as $K_{\mu \nu}$ in the term $A^{\mu}K_{\mu \nu}A^{\nu}$ of the lagrangian density is related to causal propagator (for a vector field). I mean, ...
2
votes
1answer
56 views

Deriving the path integral from the time-slice approach for a general hamiltonian

I am reading lecture notes Deriving the Path Integral by A. Wipf (pdf) trying to better understand how to derive the path integral. Specifically, equation (2.27) is: $$ K(t,q',q)=\int d\omega_1... d\...