Low dimensional anticommutation example

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In summary, the conversation discusses examples of matrices that satisfy the relation a_i a_j^{\dagger} + a_j^{\dagger} a_i = \delta_{ij}. The conversation also explores the use of creation and annihilation operators for fermions and the possibility of generalizing for multiple Fourier modes. The necessity of anticommutativity for fermion systems is also questioned.
  • #1
jostpuur
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Can somebody give an example of two matrices [itex]a_1[/itex] and [itex]a_2[/itex] which would satisfy the relation

[tex]
a_i a_j^{\dagger} + a_j^{\dagger} a_i = \delta_{ij}
[/tex]

I know that

[tex]
\left(\begin{array}{cc}
0 & 1 \\ 0 & 0 \\
\end{array}\right)
\left(\begin{array}{cc}
0 & 0 \\ 1 & 0 \\
\end{array}\right)
+ \left(\begin{array}{cc}
0 & 0 \\ 1 & 0 \\
\end{array}\right)
\left(\begin{array}{cc}
0 & 1 \\ 0 & 0 \\
\end{array}\right)
= 1
[/tex]

but I found myself unable to modify this for the index [itex]i\in\{1,2\}[/itex].
 
Last edited:
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  • #2
One obvious example is creation and annihilation operators for electrons. These operators act in the Fock space, which is a direct sum of N-electron Hilbert spaces, where N runs from 0 to infinity. So, the corresponding matrices are infinite-dimensional.

Eugene.
 
  • #3
Raising and lowering operators for fermions were my original motivation for this problem. I didn't feel comfortable by the way books only give the properties of the operators, but I also want to see how these operators could be defined. I decided to make the problem simpler by allowing "only two Fourier modes" for the particles.

My first idea was that I should use four dimensional vector space, denote the basis vectors as [itex]|00\rangle[/itex], [itex]|01\rangle[/itex], [itex]|10\rangle[/itex], [itex]|11\rangle[/itex], and then define [itex]a_1[/itex] so that it maps [itex]|01\rangle\mapsto |00\rangle[/itex] and [itex]|11\rangle\mapsto |10\rangle[/itex], and [itex]a_2[/itex] so that it maps [itex]|10\rangle\mapsto |00\rangle[/itex] and [itex]|11\rangle\mapsto |01\rangle[/itex]. With the following notation choice:

[tex]
|00\rangle = \left(\begin{array}{c}
1 \\ 0 \\ 0 \\ 0 \\
\end{array}\right),\quad
|01\rangle = \left(\begin{array}{c}
0 \\ 1 \\ 0 \\ 0 \\
\end{array}\right),\quad
|10\rangle = \left(\begin{array}{c}
0 \\ 0 \\ 1 \\ 0 \\
\end{array}\right),\quad
|11\rangle = \left(\begin{array}{c}
0 \\ 0 \\ 0 \\ 1 \\
\end{array}\right)
[/tex]

the operators would be

[tex]
a_1 = \left(\begin{array}{cccc}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
\end{array}\right),\quad\quad
a_2 = \left(\begin{array}{cccc}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}\right)
[/tex]

But this does not work as wanted. These work like this:

[tex]
a_1 a_2^{\dagger} + a_2^{\dagger} a_1 = \left(\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}\right)
[/tex]

Because [tex]a_1 a_2^{\dagger}[/tex] maps [itex]|01\rangle\mapsto |11\rangle\mapsto |10\rangle[/itex], and [tex]a_2^{\dagger} a_1[/tex] maps [itex]|01\rangle\mapsto |00\rangle\mapsto |10\rangle[/itex].
 
  • #4
I was informed elsewhere, that matrices

[tex]
a_1 = \frac{1}{\sqrt{2}}\left(\begin{array}{cc}
1 & 0 \\ 0 & -1 \\
\end{array}\right),\quad\quad
a_2 = \frac{1}{\sqrt{2}}\left(\begin{array}{cc}
0 & 1 \\ 1 & 0 \\
\end{array}\right)
[/tex]

satisfy the property I was asking for. This did not solve my problem though, because it only made me realize that I should have also demanded that [itex]a_1 a_1 = 0[/itex] and [itex]a_2 a_2 = 0[/itex] in the beginning.
 
  • #5
Well, using your 4-D vector space from post #3, I think the following matrices fit the bill:

[tex]
\left(\begin{array}{cccc}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
\end{array}\right),\quad\quad

and

\left(\begin{array}{cccc}
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
\end{array}\right)
[/tex]

but they are adjoints of each other, so perhaps that is not what you are looking for. I guess it's just a block diagonal generalization of your original 2x2 case, now that I look at it.
 
  • #6
That's right. I'm surprised to notice only now, that the matrices

[tex]
a_1 = \left(\begin{array}{cc}
0 & 1 \\ 0 & 0 \\
\end{array}\right),\quad\quad
a_2 = \left(\begin{array}{cc}
0 & 0 \\ 1 & 0 \\
\end{array}\right)
[/tex]

answer my original question

But there still seems to be something that doesn't make sense in this. Perhaps I'll need to come up with a better formulation for the problem...
 
  • #7
this is one example of dirac algebra, and can be generalized to clifford algebra.
 
  • #8
I have now understood that not every non-zero number has to be one.

So I want

[tex]a_1 |00\rangle = 0[/tex]
[tex]a_1 |01\rangle \propto |00\rangle[/tex]
[tex]a_1 |10\rangle = 0[/tex]
[tex]a_1 |11\rangle \propto |10\rangle[/tex]

[tex]a_2 |00\rangle = 0[/tex]
[tex]a_2 |01\rangle = 0[/tex]
[tex]a_2 |10\rangle \propto |00\rangle[/tex]
[tex]a_2 |11\rangle \propto |01\rangle[/tex]

and

[tex]a_ia_j^{\dagger} + a_j^{\dagger}a_i = \delta_{ij}[/tex]

If I choose any four complex numbers [itex]\alpha,\beta,\gamma,\delta[/itex] such that [itex]|\alpha|=|\beta|=|\gamma|=|\delta|=1[/itex] and [itex] \beta\delta^* + \gamma^*\alpha = 0[/itex], then the desired operators can be defined as

[tex]
a_1 = \left(\begin{array}{cccc}
0 & \alpha & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & \beta \\
0 & 0 & 0 & 0 \\
\end{array}\right),\quad\quad\quad
a_2 = \left(\begin{array}{cccc}
0 & 0 & \gamma & 0 \\
0 & 0 & 0 & \delta \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}\right)
[/tex]

For example [itex](\alpha,\beta,\gamma,\delta)=(1,1,1,-1)[/itex] is one possibility.

But there is something inelegant about this. It seems laborous to try to generalise this for three Fourier modes next. Would I need to solve again even more equations then?
 
Last edited:
  • #9
Suppose we have a fermion system with N Fourier modes. Then for [itex]i\neq j[/itex] the raising operator [tex]a_j^{\dagger}[/tex] and lowering operator [tex]a_i[/tex] will commute modulo some coefficient, so that

[tex]
a_i a_j^{\dagger} |\delta_1\delta_2 \cdots \underset{(i)}{1} \cdots \underset{(j)}{0} \cdots \delta_N\rangle
\propto |\delta_1\delta_2 \cdots \underset{(i)}{0} \cdots \underset{(j)}{1} \cdots \delta_N\rangle
[/tex]

[tex]
a_j^{\dagger} a_i|\delta_1\delta_2 \cdots \underset{(i)}{1} \cdots \underset{(j)}{0} \cdots \delta_N\rangle
\propto |\delta_1\delta_2 \cdots \underset{(i)}{0} \cdots \underset{(j)}{1} \cdots \delta_N\rangle
[/tex]

It seems, that we can always choose these coefficients so that [tex]a_i a_j^{\dagger} + a_j^{\dagger} a_i = 0[/tex] (when [itex]i\neq j[/itex]).

Why do physicist insist on getting the anticommutativity property holding? If the crucial property of fermion system is that each Fourier mode always has only one excitation state, the anticommutativity of raising and lowering operators does not seem to be inherent to the system. We might as well have fermion systems without anticommuting raising and lowering operators too, right?
 

Related to Low dimensional anticommutation example

1. What is a low dimensional anticommutation example?

A low dimensional anticommutation example is a mathematical model used to illustrate the concept of anticommutation in a system with a low number of dimensions. It involves the use of matrices to represent the different dimensions and their corresponding anticommutative properties.

2. How is anticommutation different from commutation?

Commutation and anticommutation are both mathematical operations that involve the swapping of variables or elements. However, in commutation, the order of the elements does not matter, while in anticommutation, the order does matter and can result in a sign change.

3. What are some real-world examples of anticommutation?

Some real-world examples of anticommutation include the Pauli exclusion principle in quantum mechanics, where the exchange of two identical particles can result in a change in the overall wave function; and the cross product of vectors in physics, where the order of the vectors matters and can result in a change in the direction of the resulting vector.

4. How is anticommutation used in quantum computing?

Anticommutation plays a crucial role in quantum computing, as it is used to represent and manipulate quantum states, which are described by multi-dimensional vectors known as qubits. The anticommutative properties of qubits allow for complex calculations and algorithms to be performed, making quantum computing more powerful than classical computing.

5. What are the applications of studying low dimensional anticommutation?

Studying low dimensional anticommutation can have applications in various fields such as physics, mathematics, and computer science. It can help in understanding complex systems and phenomena, developing new mathematical models and algorithms, and advancing technologies such as quantum computing and encryption.

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