# Questions tagged [commutator]

A mathematical construct quantifying the difference in effect of applying two operators in two alternate successions. It is the defining product of a Lie algebra, the efficient underlying description of Lie groups, of use in several areas of physics, most notably quantum field theory.

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### What is the physical meaning of commutators in quantum mechanics?

This is a question I've been asked several times by students and I tend to have a hard time phrasing it in terms they can understand. This is a natural question to ask and it is not usually well ...
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### Trace of a commutator is zero - but what about the commutator of $x$ and $p$?

Operators can be cyclically interchanged inside a trace: $${\rm Tr} (AB)~=~{\rm Tr} (BA).$$ This means the trace of a commutator of any two operators is zero: $${\rm Tr} ([A,B])~=~0.$$ But what about ...
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### What is the mistake in calculating such a commutator? [duplicate]

$B$ is an Hermitian operator in Hilbert space, and $|b\rangle$ is the eigenstate of $B$. We can have $[A, B] = 1$ where A is arbitary operator. Then we can calculate as below: \begin{align} &\...
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### Quantizing a complex Klein-Gordon Field: Why are there two types of excitations?

In most references I've seen (see, for example, Peskin and Schroeder problem 2.2, or section 2.5 here), one constructs the field operator $\hat{\phi}$ for the complex Klein-Gordon field as follows: ...
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### What exactly is the connection between the Jacobi and Bianchi identities?

While reviewing some basic field theory, I once again encountered the Bianchi identity (in the context of electromagnetism). It can be written as $$\partial_{[\lambda}\partial_{[\mu}A_{\nu]]}=0.$$ ...
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### Renormalization and canonical commutation relations

My question is whether canonical commutation relations hold for renormalized quantum fields. Below I show reasoning which caused by doubts. Consider a relativistic scalar QFT. We have spectral ...
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### Commutator algebra in exponents

Considering $X$ and $Y$ such that $[X,Y]=\lambda$, which is complex, and $\mu$ is another complex number, prove: $$e^{\mu(X+Y)}=e^{\mu X} e^{\mu Y} e^{-\mu^2\lambda/2}$$ My attempt (so far) is: ...
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### Does the commutator of anything with itself not vanish?

In a quantum mechanics exam one question was to write the commutator of a couple of operators. Everybody got points taken away since they did not write $[Q_i, Q_i] = 0$ for all the operators $Q_i$ in ...
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### Binomial expansion of non-commutative operators

I would like to determine the general expansion of $$(\hat{A}+\hat{B})^n,$$ where $[\hat{A},\hat{B}]\neq 0$, i.e. $\hat{A}$ and $\hat{B}$ are two generally non-commutative operators. How could I ...
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### Quantum mechanics with multiple values of $\hbar$

The quantity $\hbar$ appears in quantum mechanics by the canonical commutation relation $$[x, p] = i \hbar.$$ Would it be sensible to quantize different conjugate variable pairs with different values ...
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### Derive canonical commutation relations from Schwingers principle

The book of Dyson "Quantum-Field-Theory" states in section 4.4 that one can derive canonical commutation relations from Schwingers quantum action principle. However, doesn't give a calculation for the ...
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### Commutator of Lorentz boost generators : visual interpretation

I have always struggled to visualize the correctness of the commutation relation for the generators of the boost in the Lorentz group. We have $$[K_i,K_j] = i \epsilon_{ijk} L_k$$ I fail to picture ...
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### Proof of Canonical Commutation Relation (CCR)

I am not sure how $QP-PQ =i\hbar$ where $P$ represent momentum and $Q$ represent position. $Q$ and $P$ are matrices. The question would be, how can $Q$ and $P$ be formulated as a matrix? Also, what is ...
Let's say we have two modes, with the following labeling of occupation number states: $\lvert \Psi \rangle = \begin{pmatrix} 0,0 \\ 0,1 \\ 1,0 \\ 1,1 \end{pmatrix}$ An example of (what I assume to ...