Is chirality operator defined in odd dimensions?

In summary, it is not possible to define the chirality operator in odd dimensions, including 3+1 dimensions. In even dimensions, a generalized chirality operator can be defined, but it will always be a multiple of the unit matrix. Therefore, it is not a useful operator in 3+1 dimensions.
  • #1
arroy_0205
129
0
As far as I remember, I heard from someone that the matrix
[tex]
\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3
[/tex]
also known as the chirality operator in 3+1 dimension is not defined in odd dimensions. I do not understand why that should be the case. Suppose I am in the 4+1 dimension and I choose one more gamma matrix suitably to close the Clifford algebra in five dimensions and then define analogously to the 4 dimensional case, the operator
[tex]
\Gamma^c=i\gamma^0\gamma^1\gamma^2\gamma^3\gamma^4
[/tex]
as my chirality operator. Will that be a mistake?
What about six dimensions? Can I define my chirality operator by multiplying the required no. of basic gamma matrices in that dimension?
 
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  • #2
The STA basis vectors, which is also a clifford algebra, (http://en.wikipedia.org/wiki/Space-time_algebra), I believe map to the matrix respresentation you are using. With that basis, the value:

[tex]
i = \gamma_0\gamma_1\gamma_2\gamma_3 = -\gamma^0\gamma^1\gamma^2\gamma^3
[/tex]

where i is the usual pseudoscalar for the space, and i^2 = -1. Not knowing exactly what your matrix representation is, I'd guess you have:

[tex]
i\gamma^0\gamma^1\gamma^2\gamma^3 = -\gamma^0
[/tex].

If that's the case, then it doesn't appear to me that this is a very useful operator in 3+1 dimensions.

Before thinking about the 4+1 case, what matrixes are you using for [itex]\gamma^{\mu}[/itex] in the 3+1 dimensional case?
 
  • #3
arroy_0205 said:
As far as I remember, I heard from someone that the matrix
[tex]
\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3
[/tex]
also known as the chirality operator in 3+1 dimension is not defined in odd dimensions. I do not understand why that should be the case. Suppose I am in the 4+1 dimension and I choose one more gamma matrix suitably to close the Clifford algebra in five dimensions and then define analogously to the 4 dimensional case, the operator
[tex]
\Gamma^c=i\gamma^0\gamma^1\gamma^2\gamma^3\gamma^4
[/tex]
as my chirality operator. Will that be a mistake?
What about six dimensions? Can I define my chirality operator by multiplying the required no. of basic gamma matrices in that dimension

In any dimensions, (n = 2p, or n = 2p + 1), a "generilised [itex]\gamma_{5}[/itex]" can be defined by

[tex]\Gamma_{n + 1} \equiv \Gamma_{1}\Gamma_{2}...\Gamma_{n}[/tex]

From the algebra

[tex]\{\Gamma_{a}, \Gamma_{b}\} = 2 \eta_{ab} \ I[/tex]

it follows that [itex]\Gamma_{n+1}[/itex] anticommutes with all [itex]\Gamma_{a}[/itex] for even dimensions ( n = 2p ), i.e.,

[tex]\{\Gamma_{n+1}, \Gamma_{a}\} = 0 \ \ \forall a = 1,2,..,2p[/tex]

but, for odd dimensions (n = 2p +1), it COMMUTES with all [itex]\Gamma_{a}[/itex], i.e.,

[tex][\Gamma_{n+1},\Gamma_{a}] = 0, \ \ \forall a = 1,2,..,2p+1[/tex]

Therefore in odd dimensions, by Schur's lemma, [itex]\Gamma_{n+1}[/itex] is a multiple of the unit matrix. This fact is valid in any representation you choose for the gamma matrices.

regards

sam
 

1. What is chirality operator?

The chirality operator is a mathematical tool used in physics to describe the handedness or "handedness" of a particle. It is commonly used in quantum mechanics to describe the spin of particles and their interactions with other particles.

2. How is chirality operator defined?

The chirality operator is defined as a matrix or operator that acts on the wavefunction of a particle. In odd dimensions, it can be represented by a gamma matrix, which is a matrix with a specific set of properties that allow it to describe the spin and handedness of particles.

3. Is the chirality operator defined in all dimensions?

No, the chirality operator is only defined in odd dimensions. In even dimensions, there is no unique representation of the chirality operator and it cannot be used to describe the spin of particles.

4. What is the significance of the chirality operator in physics?

The chirality operator is significant because it allows us to describe the behavior of particles in terms of their spin and handedness. This is important in understanding the properties and interactions of particles, and has applications in fields such as quantum mechanics and particle physics.

5. How is the chirality operator used in experiments?

The chirality operator is used in experiments by measuring the spin and handedness of particles and their interactions with other particles. This can provide valuable information about the properties of particles and can help validate theories in physics.

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