# 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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### Mutual or same set of eigenfunctions if two operators commute

If two operators commute, do they have "a mutual set of eigenfunctions", or "the same set of eigenfunctions"? My quantum chemistry book uses these as if they are interchangeable, but they do not seem ...
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### Commutator with exponential $[A, \exp(B)]$

How can I tell if $A$ and $\exp(B)$ commute? For $[A, B]$ it's simply $AB-BA$ and for $[\exp(A), \exp(B)]$ I think it'd be $\exp(A)\exp(B) - \exp(B)\exp(A) = \exp(A+B) - \exp(B+A) = 0$. Update: it's ...
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### Symmetry in quantum mechanics

My professor told us that in quantum mechanics a transformation is a symmetry transformation if $$UH(\psi) = HU(\psi)$$ Can you give me an easy explanation for this definition?
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### Fundamental Commutation Relations in Quantum Mechanics

I am trying to compile a list of fundamental commutation relations involving position, linear momentum, total angular momentum, orbital angular momentum, and spin angular momentum. Here is what I have ...
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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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### 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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### 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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### 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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In some books, a proof that if two self-adjoint operators $A$ and $B$ share a common eigenbasis $\{\phi_n\}$, then they commute is given as follows : For any $\phi_n$, $$AB\ \phi_n = a_n\ b_n\ ... 2answers 1k views ### Dilation operator in CFT viewed as 'hamiltonian'? From the commutation relations for the conformal Lie algebra, we may infer that the dilation operator plays the same role as the Hamiltonian in CFTs. The appropriate commutation relations are [D,P_{... 4answers 1k views ### Fermions, different species and (anti-)commutation rules My question is straightforward: Do fermionic operators associated to different species commute or anticommute? Even if these operators have different quantum numbers? How can one prove this fact in a ... 1answer 554 views ### Why don't we use Hamilton-Jacobi method in QM? In classical mechanics, we usually try to find a set of coordinates by Hamilton-Jacobi method to transform the Hamiltonian to zero such that the coordinates are conservations. However, we never try ... 1answer 523 views ### Theories with non-vanishing commutators outside the lightcone I'm reading Weinberg's new book on Quantum Mechanics, and in Chapter 8.7 "Time-Dependent Perturbation Theory" he derives the usual Dyson series for the S matrix when the interaction Hamiltonian V_I(... 2answers 489 views ### How to replace T-product with retarded commutator in LSZ formula? I am reading Itzykson and Zuber's Quantum Field Theory book, and am unable to understand a step that is made on page 246: Here, they consider the elastic scattering of particle A off particle B: ... 2answers 706 views ### How to know if a set of commuting observables is complete? We define a complete set of commuting observables as a set of observables \{A_1,\ldots, A_n\} such that: \left[A_i, A_j\right]=0, for every 1\leq i,~j \leq n; If a_1,\ldots, a_n are ... 3answers 3k views ### Canonical Commutation Relations Is it logically sound to accept the canonical commutation relation (CCR)$$[x,p]~=~i\hbar$$as a postulate of quantum mechanics? Or is it more correct to derive it given some form for p in the ... 4answers 13k views ### Commutator of Dirac gamma matrices Quick question...For some reason I'm having trouble finding an identity or discussion for the commutator of the gamma matrices at the moment...i.e$$\gamma^u\gamma^v-\gamma^v \gamma^u$$but I am not ... 3answers 1k views ### Commutator with a square root How to find the commutator [a, \sqrt{a^\dagger a}]? Here a is a usual bosonic annihilation operator, and [a, a^\dagger] = 1. The first thing I tried is$$ [x,A] = [x, \sqrt{A}]\sqrt{A} + \sqrt{...
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 ...