Linked Questions

22 votes
1 answer
10k views

Why/How is this Wick's theorem?

Let $\phi$ be a scalar field and then I see the following expression (1) for the square of the normal ordered version of $\phi^2(x)$. \begin{align} T(:\phi^2(x)::\phi^2(0):) &= 2 \langle 0|T(\...
user6818's user avatar
  • 4,709
18 votes
5 answers
806 views

Can I swap quantum mechanical ground state for some classical trajectory distribution and have it sit still after the swap?

Suppose that I have a single massive quantum mechanical particle in $d$ dimensions ($1\leq d\leq3$), under the action of a well-behaved potential $V(\mathbf r)$, and that I let it settle on the ground ...
Emilio Pisanty's user avatar
7 votes
1 answer
4k views

Normal Order of Normal Order

In the first volume of Polchinski page 39 we can read a compact formula to perform normal-order for bosonic fields $$ :\cal F:~=~\underbrace{\exp\left\{\frac{α'}{4}∫\mathrm{d}^2z\mathrm{d}^2w\log|z-w|^...
MaPo's user avatar
  • 1,556
9 votes
2 answers
2k views

How to promote algebraic expressions to operators in quantum mechanics?

Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription $$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} $...
guru's user avatar
  • 933
14 votes
2 answers
1k views

Motivation for Wigner phase space distribution

Most sources say that Wigner distribution acts like a joint phase-space distribution in quantum mechanics and this is justified by the formula $$\int_{\mathbb{R}^6}w(x,p)a(x,p)dxdp= \langle \psi|\...
user avatar
11 votes
1 answer
1k views

In quantum mechanics, how exactly do we associate Hermitian operators to classical observables? [duplicate]

In a first course on quantum mechanics, everybody learns some version of the following statement: Postulate: To every classical observable $A$ of a physical system, there corresponds a Hermitian ...
David Zhang's user avatar
4 votes
2 answers
2k views

Examples of Weyl transforms of nontrivial operators

I've been able to find examples of Weyl transforms of operators like $\hat{x}$,$\hat{p}$, and $\hat{1}$, but not anything more complicated. Are there derivations of the Weyl transforms of more ...
Dan's user avatar
  • 5,765
3 votes
1 answer
1k views

Can I Weyl-order the following Hamiltonian?

I am trying to perform a path integral but I am having trouble with the Weyl ordering of my Hamiltonian. The Lagrangian of the system in question is $$L~=~\frac{1}{2}f(q)\dot{q}^2,$$ where $f(q)$ ...
Yossarian's user avatar
  • 6,137
5 votes
1 answer
1k views

Weyl and Normal ordering in QFT

In my QFT course, if I understood well, we define the normal ordering as the way to quantize a system where we put the creation operators at the left and the annihilation ones at the right. For ...
StarBucK's user avatar
  • 1,560
5 votes
2 answers
557 views

Why any operator is specified by this characteristic function?

On the paper "Tutorial Notes on One-Party and Two-Party Gaussian States", arXiv:quant-ph/0307196, the author states on section 2: Any operator referring to a harmonic oscillator — position ...
Gold's user avatar
  • 37.4k
3 votes
2 answers
649 views

Wigner transform & convolution

I'm trying to understand the gradient expansion within the Keldysh formalism. In particular, I am reading "Quantum Field Theory of Non-equilibrium States" by J. Rammer, section 7.2, ...
surrutiaquir's user avatar
8 votes
2 answers
258 views

How do we define the Heisenberg picture within functorial/path integral QFT?

In the functorial approach to QFT, each Cauchy surface $\Sigma$ has an associated Hilbert space $\mathcal{H}_\Sigma$, and each pair of Cauchy surfaces $\Sigma,\Sigma'$ has an associated unitary $U_{\...
nodumbquestions's user avatar
2 votes
2 answers
210 views

What does it mean for an operator to depend on position or momentum?

While trying to provide an answer to this question, I got confused with something which I think might be the root of the problem. In the paper the OP was reading, the author writes $$\frac{d\hat{A}}{...
Lourenco Entrudo's user avatar
1 vote
1 answer
257 views

Operator Ordering Conventions and Symmetry

Quantization procedures may need operator ordering conventions to avoid ambiguity. In classical theories, classical observables are often described by smooth functions, so the order of observable ...
leob's user avatar
  • 569
3 votes
1 answer
97 views

Order of product of $x$ and $p$ while deriving Hamiltonian from a Lagrangian in Quantum Mechanics

Everyone who has taken a course in Quantum Mechanics has at some point derived a quantum Hamiltonian from a Lagrangian. However, I can't seem to find any reference on the topic. My question is ...
SlothForeva's user avatar

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