All Questions
Tagged with anticommutator commutator
66
questions
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Does the anticommutator of two spinors affect the transpose of their product?
My lecture notes claim that for an anticommutation relation
$$[ \psi_{\mu}(\bf{x},t),{\psi_{{\nu}}^{*}}(\bf{y},t)] = \delta_{\mu \nu} \delta^3(\bf{x}-\bf{y})$$
between two spinors, the transpose of ...
- 337
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0
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56
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How to generalize the (anti)commutation for spacelike separation to more than $2$ field operators?
Let $\phi_1$ and $\phi_2$ be quantum field operators of certain spin on $\mathbb{R}^4$. Then, the principle of locality dictates that if $x$ and $y$ are space-like separated, we have
\begin{equation}
\...
- 1,669
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111
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Quantization of an Interacting Field Theory
The procedure to quantize free field theories is imposing a commutation/anticommutation relation with the field and its conjugate momentum, as $$\mathcal L = i\bar\psi\gamma^\mu\partial_\mu\psi\...
- 21
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Why does $[Q,P]=i\hbar$ work for fermion? Shouldn't fermion satisfy anticommuting relation?
For hydrogen, we use $[Q,P]=i\hbar$ for electron, which is a fermion. Does it have a deeper reason such as that we're really considering the proton + electron system, which might be of bosonic nature?
- 619
4
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Why is commutation bracket used instead of anti-commutation bracket on page 61 of Peskin QFT?
Peskin&Schroeder was performing a trick where they used
$$J_za^{s\dagger}_0|0\rangle=[J_z,a^{s\dagger}_0]|0\rangle\tag{p.61}$$ and claimed that the only non-zero term in this commutator would be ...
- 838
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88
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Help with commutator algebra with fermionic operators
I am struggling to understand how the following is true for the fermionic creation/annihilation operators $a^\dagger, a$: $$[a^\dagger a, a]=-a$$
If someone could walk me through the math derivation ...
- 101
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125
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Why do we only consider commutators and anticommutators in QFT?
While studying canonical quantization in QFT, I observed that we quantize fields either by a commutation or an anticommutation relation
\begin{equation}
[\phi(x), \phi(y)]_\pm := \phi(x) \phi(y) \pm \...
- 1,588
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1
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79
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What is the motivation for the Self-Dual Canonical Anticommutation Relation (CAR) algebra?
What exactly is the motivation for the use of the Self-Dual Canonical Anticommutation Relation (CAR) algebra in the context of infinite lattice systems? Why not remain with the CAR algebra, as both ...
- 71
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Why can the commutator of a general expression be replaced by the anti-commutator in the $bc$ CFT theory?
Polchinski states in his equation 2.6.14 (in his book String Theory Vol. 1, Introduction to the Bosonic String) that for charges $Q_1$ and $Q_2$ the following equation holds, where $j_i$ is the ...
- 938
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238
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Commutation of kinetic energy operator with Hamiltonian
I am basically trying to calculate current energy operator $\hat{\mathbf{J}}_E(\mathbf{r})$ by using Heisenberg equation of motion as
$$
-\nabla\cdot \hat{\mathbf{J}}_E(\mathbf{r})=\frac{i}{\hbar}[H,\...
- 831
3
votes
1
answer
98
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Contour integral for commutator of fermionic fields
Suppose we have primary fields $A$ and $B$ which have the OPE,
$$A(z) B(w) = \frac{1}{z-w} = -B(z)A(w), \quad |z| > |w|,\tag{1}$$
so they have fermionic statistics. Now I was curious how this would ...
- 721
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128
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Can Hadamard's formula be used for fermionic operators?
Can I use this special case of Hadamard's formula
$$e^\hat B \hat A e^{-\hat B}= A + [B,A]+\frac{1}{2!}[B, [B,A]] + \dots$$
for fermionic operators?
Suppose I have fermionic operators that obey ...
- 179
1
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1
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122
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Can the operator field Dirac equation be expressed as Heisenberg's equation?
The Dirac equation of the operator spinor field is:
$$(i\gamma ^{\mu}\partial _{\mu} -m)\psi =0$$
where $\psi$ is interpreted to be a quantum field.
I'm wondering, can this be derived from the ...
- 6,355
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0
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124
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Is there a Stone-von-Neumann theorem-like result for the canonical anti-commutation relations (CAR)?
The canonical commutation relation (CCR)
$$[\phi(x), \pi(y)] = i\hbar\delta(x-y)$$
is kind of the key to basically any bosonic quantum theory. This is due to many different remarkable properties: By ...
- 6,763
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1
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162
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Which of these commutation relations are correct? [closed]
I saw, in two different references, the following two commutation relations for the fermionic field operator:
and
which one of them is correct?
1 "Stefanucci, Gianluca, and Robert Van Leeuwen. ...
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