Commutation relations for bosons and fermions

In summary, the field operators for free bosons and fermions satisfy different commutation relations at equal times, where fermions cannot be created at the same position and bosons prefer to be in the same state. These commutation relations reflect the Pauli exclusion principle.
  • #1
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For the free boson, the field operators satisfies the commutation relation,

$${\varphi}_{x'}{\varphi}_{x} - {\varphi}_{x}{\varphi}_{x'} = 0$$ at equal times.

While the fermions satisfies,

$${\psi}_{x'}{\psi}_{x} + {\psi}_{x}{\psi}_{x'} = 0$$ at equal times.

I interpret ##{\varphi}_{x}## and ##{\psi}_{x'}## as creating a boson and a fermion at position x and x' respectively.

But what is the physical interpretation of the commutations relations? I'm trying to relate it to the fact that fermions changes sign when any two fermions are interchanged, while bosons do not.
 
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  • #2
You can't create two fermions at the same place at the same time (but I bet you already knew that). ##\psi (x) \psi (x) + \psi(x)\psi(x)=0## so ##\psi(x)\psi(x)=0##

Bosons, on the other hand, "like" to be in the same state.
 
  • #3
Gene Naden said:
You can't create two fermions at the same place at the same time (but I bet you already knew that). ##\psi (x) \psi (x) + \psi(x)\psi(x)=0## so ##\psi(x)\psi(x)=0##

Bosons, on the other hand, "like" to be in the same state.

They are not created at the same place. Did you notice the prime on the x's`?
 
  • #4
But ##x## and ##x^\prime## are arbitrary. They can be equal. When they are equal then they are created at the same place. So that is part of the physical meaning: they cannot be created at the same place. The commutation and anti-commutation relations reflect the Pauli principle, I believe.

Anyway, the full relation is ##\psi(x)\psi(x^\prime)+\psi(x^\prime)\psi(x)=\delta(x-x^\prime)## so that reflects equality of ##x## and ##x^\prime##
 

1. What are commutation relations for bosons and fermions?

Commutation relations are mathematical representations of the relationships between operators that describe the properties of particles. For bosons, the commutation relation is [a, a†] = 1, while for fermions, it is {b, b†} = 1. These relations play a crucial role in understanding the behavior of quantum particles.

2. How are commutation relations for bosons and fermions different?

The main difference between commutation relations for bosons and fermions lies in the type of particles they describe. Bosons are particles with integer spin, such as photons, while fermions have half-integer spin, such as electrons. This difference leads to variations in their commutation relations, which have important consequences for their behavior.

3. What do the symbols a and b represent in commutation relations for bosons and fermions?

The symbols a and b represent creation and annihilation operators, respectively. These operators act on the states of particles and determine their properties, such as energy and momentum. In commutation relations, they are represented by a and b for bosons and fermions, respectively.

4. How do commutation relations for bosons and fermions affect particle statistics?

Commutation relations play a crucial role in determining the statistical behavior of particles. For bosons, the commutation relation results in symmetric wave functions, leading to Bose-Einstein statistics. On the other hand, for fermions, the anti-commutation relation leads to antisymmetric wave functions, which give rise to Fermi-Dirac statistics.

5. What are the applications of commutation relations for bosons and fermions?

Commutation relations have important applications in various fields of physics, including quantum mechanics, quantum field theory, and condensed matter physics. They help in understanding the properties and behavior of different types of particles and are also used in developing quantum technologies, such as quantum computing and quantum cryptography.

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