Linked Questions

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0answers
56 views

Commutation relations in QFT [duplicate]

So I have just started learning QFT. So you take a classical field and turn the degrees of freedom into operators. All fine, just like normal quantum. However I am confused about the commutation ...
8
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2answers
6k views

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 ...
13
votes
4answers
3k 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 ...
4
votes
3answers
9k views

Fermionic anti-commutation relations

For Pauli's exclusion principle to be followed by fermions, we need these anti-commutators $$[a_{\lambda},a_{\lambda}]_+=0 $$ and $$[a_{\lambda}^{\dagger},a_{\lambda}^{\dagger}]_+=0 $$ Then $$n_{\...
20
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1answer
3k views

Classical Fermion and Grassmann number

In the theory of relativistic wave equations, we derive the Dirac equation and Klein-Gordon equation by using representation theory of Poincare algebra. For example, in this paper http://arxiv.org/...
13
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1answer
3k views

Making sense of the canonical anti-commutation relations for Dirac spinors

When doing scalar QFT one typically imposes the famous 'canonical commutation relations' on the field and canonical momentum: $$[\phi(\vec x),\pi(\vec y)]=i\delta^3 (\vec x-\vec y)$$ at equal times ($...
8
votes
1answer
2k views

What goes wrong when one tries to quantize a scalar field with Fermi statistics?

At the end of section 9 on page 49 of Dirac's 1966 "Lectures on Quantum Field Theory" he says that if we quantize a real scalar field according to Fermi statistics [i.e., if we impose Canonical ...
5
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1answer
2k views

The implication of anti-commutation relations in quantum mechanics

All the textbooks I saw are very clear about the implications of commutating operators in quantum mechanics. However, much less is said about anti-commutation relations. Does it have a general ...
1
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1answer
2k views

Integral spin and half integral spin

I am reading a book (Laudau and Lifshitz, Vol. 4, page 94) and it derived why spin-0 should obey Bose quantization and spin-1/2 should obey Fermi Quantization. Then it says, all integral spin ...
5
votes
1answer
1k views

In QFT, why do fermions have to anticommute in order to insure causality?

I have seen this question and I believe I understand the answer to it. However, AFAIK, only for bosons the causality condition is a vanishing commutator. For fermions we expect the anticommutator $[\...
5
votes
2answers
1k views

Imposing anti-commutation relations on fermionic quasi-particles

In many theories of CMT, we assume the nature of quasi-particles (without giving proper justifications). For example, we assume nature of quasi-particles to be fermionic in case of a interacting ...
2
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1answer
638 views

Why must fermion fields anticommute and bosons commute?

Fermion fields must satisfy anticommutation relation. But why? I know that unless they anti-commute the Pauli exclusion principle cannot be satisfied. But is there some other deeper/fundamental ...
3
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1answer
487 views

What's the reasoning behind propagators definitions (specifically fermionic propagators)

I'm studying QFT by David Tong's lecture notes. When he discusses causility with real scalar fields, he defines the propagator as $$D(x-y)=\left\langle0\right|\phi(x)\phi(y)\left|0\right\rangle=\int\...
0
votes
1answer
532 views

Quantizing the Dirac field using commutation relations leads to an unbounded Hamiltonian?

If we were to quantize the Dirac field using commutation relations instead of anticommutation relations we would end up with the Hamiltonian $$ H = \int\frac{d^3p}{(2\pi)^3}E_p \sum_{s=1}^2 ...
1
vote
2answers
594 views

How will the (anti)commutation relation between two different fermion fields look like? [duplicate]

The anti-commutation relation between the components of a fermion field $\psi$ is given by $$[\psi _\alpha(x),\psi_\beta^\dagger(y)]_+=\delta_{\alpha\beta}\delta^{(3)}(\textbf{x}-\textbf{y}).$$ In ...

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