Linked Questions
22
votes
1
answer
10k
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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(\...
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 ...
7
votes
1
answer
4k
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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
$$
:\cal F:~=~\underbrace{\exp\left\{\frac{α'}{4}∫\mathrm{d}^2z\mathrm{d}^2w\log|z-w|^...
9
votes
2
answers
2k
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How to promote algebraic expressions to operators in quantum mechanics?
Okay, I know that in quantum mechanics the quantum observable is obtained from the classical observable by the prescription
$$ X \rightarrow x,\quad P \rightarrow -i\hbar\frac{\partial}{\partial x} $...
14
votes
2
answers
1k
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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
$$\int_{\mathbb{R}^6}w(x,p)a(x,p)dxdp= \langle \psi|\...
11
votes
1
answer
1k
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In quantum mechanics, how exactly do we associate Hermitian operators to classical observables? [duplicate]
In a first course on quantum mechanics, everybody learns some version of the following statement:
Postulate: To every classical observable $A$ of a physical system, there corresponds a Hermitian ...
4
votes
2
answers
2k
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Examples of Weyl transforms of nontrivial operators
I've been able to find examples of Weyl transforms of operators like $\hat{x}$,$\hat{p}$, and $\hat{1}$, but not anything more complicated. Are there derivations of the Weyl transforms of more ...
3
votes
1
answer
1k
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Can I Weyl-order the following Hamiltonian?
I am trying to perform a path integral but I am having trouble with the Weyl ordering of my Hamiltonian.
The Lagrangian of the system in question is
$$L~=~\frac{1}{2}f(q)\dot{q}^2,$$
where $f(q)$ ...
5
votes
1
answer
1k
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Weyl and Normal ordering in QFT
In my QFT course, if I understood well, we define the normal ordering as the way to quantize a system where we put the creation operators at the left and the annihilation ones at the right.
For ...
5
votes
2
answers
557
views
Why any operator is specified by this characteristic function?
On the paper "Tutorial Notes on One-Party and Two-Party Gaussian States", arXiv:quant-ph/0307196, the author states on section 2:
Any operator referring to a harmonic oscillator — position ...
3
votes
2
answers
649
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Wigner transform & convolution
I'm trying to understand the gradient expansion within the Keldysh formalism. In particular, I am reading "Quantum Field Theory of Non-equilibrium States" by J. Rammer, section 7.2, ...
8
votes
2
answers
258
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How do we define the Heisenberg picture within functorial/path integral QFT?
In the functorial approach to QFT, each Cauchy surface $\Sigma$ has an associated Hilbert space $\mathcal{H}_\Sigma$, and each pair of Cauchy surfaces $\Sigma,\Sigma'$ has an associated unitary $U_{\...
2
votes
2
answers
210
views
What does it mean for an operator to depend on position or momentum?
While trying to provide an answer to this question, I got confused with something which I think might be the root of the problem. In the paper the OP was reading, the author writes $$\frac{d\hat{A}}{...
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 ...
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 ...