11
votes
2answers
417 views

Why uncertainty principle is not like this?

In Griffiths' QM, he uses two inequalities (here numbered as $(1)$ and $(2)$) to prove the following general uncertainty principle: $$\sigma_A^2 \sigma_B^2\geq\left(\frac{1}{2i}\langle [\hat A ,\hat ...
4
votes
1answer
152 views

Does This Really “Prove” Spin-statistics Theorem?

In quantization of scalar field theory we impose commutation relation between the field operators by hand and similarly we impose anti-commutation relation between Dirac field operators by hand. As a ...
2
votes
3answers
176 views

Schroedinger field operators and their commutation relations

I've got several questions regarding the so called second quantization of the Schroedinger equation. My professor introduced the field operators for the Schroedinger field by simply stating them as ...
0
votes
1answer
86 views

Observables still commute even if fields only anti-commute

In Peskin & Schroeder page 56, after introducing anti commutation relations for the fields instead of commutation relations (in order to fix the negative energy problem as well as to have proper ...
4
votes
1answer
139 views

Quantizing the Dirac Field: which commutation relations are more fundamental?

When quantizing a system, what is the more (physically) fundamental commutation relation, $[q,p]$ or $[a,a^\dagger]$? (or are they completely equivalent?) For instance, in Peskin & Schroeder's ...
1
vote
1answer
155 views

A few simple questions about Grassmann numbers: commutation relations and derivatives

I'm trying to learn about Grassmann numbers from the book "Condensed Matter Field Theory" by Altland and Simons, but I am currently encountering some difficulties. I have several smaller questions ...
2
votes
1answer
68 views

Fock state and corresponding relations for continuous momentum label

In Wikipedia I found following relation for Fock state: $$ \hat {a}_i| \{n_j\}_j\rangle ~=~ \sqrt{n}_i| \{n_j-\delta_{ij}\}_j\rangle, $$ where $n_j$ refers to the number of $j$'th particles. This ...
4
votes
2answers
263 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 ...
1
vote
1answer
153 views

transformations with commutators and anticommutators that generate displacements

is well known that composition of point reflections generate pure displacements. This implies that the commutator of two point reflections will be a pure displacement. Are there similar elemental ...
14
votes
4answers
3k views

What is the connection between Poisson brackets and commutators?

The Poisson bracket is defined as: $$\{f,g\}_{PB} ~:=~ \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial ...
3
votes
1answer
719 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 ...
4
votes
2answers
649 views

Why are anticommutators needed in quantization of Dirac fields?

Why is the anticommutator actually needed in the canonical quantization of free Dirac field?
8
votes
5answers
9k views

What is the Physical Meaning of Commutation of Two Operators?

I understand the mathematics of commutation relations and anti-commutation relations, but what does it physically mean for an observable (self-adjoint operator) to commute with another observable ...