Can the usual inner product be defined on spinor space?

In summary, the conversation discusses the need to rotate and complex transpose components of Dirac spinors in order to properly multiply into SL(2,C) scalars. The question arises about the impossibility of defining the usual inner product on the complex vector space of spinors due to the non-compactness of the Lorentz group. However, it is clarified that while the generators of the spinor representation of Lorentz cannot be made all hermitian, they can still be made hermitian if they are not finite dimensional matrices. The conversation also mentions the importance of unitary representations in developing a quantum theory within special relativity.
  • #1
gentsagree
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I would like to gain a more formal mathematical understanding of a construct relating to spinors.

When I write down Dirac spinors in the Weyl basis, I see why if I multiply the adjoint (conjugate transpose) of a spinor with the original spinor I don't get a SL(2,C) scalar. It just doesn't work if we stick with the conventions of multiplication between (row and column) vectors. To clarify, here I am thinking of column vectors in spinor space as elements of a complex vector space carrying the fundamental representations of the complexified Clifford algebra, Cl(C). Rows are the complex conjugate, transposed object.

Thus one sees that one needs to "rotate" the components of the Dirac spinor (say with [itex] \gamma^{0}[/itex]), as well as complex transpose them, to be able to multiply properly into SL(2,C) scalars.

My question is the following. From the perspective of vector spaces, what goes wrong with defining the usual inner product [itex] x^{\dagger}x[/itex] on this complex vector space (i.e. the spinor space)? Does this construct have a particular name in the maths literature?

Thanks!
 
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  • #3
I can see now that all my question boils down to is that the generators of the spinor representation of Lorentz cannot be all made to be hermitian. Say we make rotations anti-hermitian and boosts hermitian. This implies the weird inner product I mention above.

Does the fact that the Lorentz group is non-compact imply in an obvious way that the generators can't be made all hermitian?

Thanks
 
  • #4
But you misunderstood the impact of non-compactness: you CAN make all its generators hermitean, but they won't be finite dimensional matrices anymore. That's the trick: you want spinors, you have to drop hermiticity for all six at the same time.

To go even further, vanhees's statement is not quite exact. The theorem involved in the representation theory of the proper Lorentz group reads:

upload_2016-2-13_0-43-58.png

That simple added there is important. It means the group shouldn't have proper invariant subgroups (which is the case for the proper Lorentz group). If the group is not simple, then the theorem won't hold.
 
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  • #5
True, I plead guilty as charged. Where is the quote of the theorem from?

Indeed, it's important to stress that there are unitary (ray) representations of the proper orthochronous Poincare group, because that implies that there exists a quantum theory in the usual sense within special relativity. You come quite far in developing it by investigating all the possible unitary representations, which goes back to Wigner's famous paper on the subject.
 
  • #6
It's the only source for this much used theorem which I could find: 2nd volume of Cornwell's book on group theory, chapter 15.
 
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  • #7
Thanks!
 

1. What is an inner product on spinor space?

An inner product on spinor space is a type of mathematical operation that takes two elements of a spinor space and produces a complex number as a result. It is a generalization of the dot product in vector spaces, and it is used to measure the angle between two spinors and to define lengths and orthogonality within spinor space.

2. How is an inner product on spinor space defined?

An inner product on spinor space is defined by a set of axioms that it must satisfy. These include properties such as linearity, symmetry, and positive definiteness. The exact definition may vary depending on the type of spinor space being used, such as complex or quaternionic spinor spaces.

3. What is the importance of inner products on spinor space?

Inner products on spinor space have many important applications in physics and mathematics. They are used in quantum mechanics to describe the state of a particle, in general relativity to define the curvature of spacetime, and in Lie algebras to study symmetries and group representations. They also have applications in other areas, such as computer graphics and signal processing.

4. How do inner products on spinor space relate to spinors?

Spinors are mathematical objects that are used to represent the intrinsic angular momentum of particles in quantum mechanics. Inner products on spinor space are used to define the inner product between two spinors, which is necessary for calculating physical quantities such as spin and magnetic moments.

5. Can inner products on spinor space be generalized to other mathematical structures?

Yes, the concept of an inner product can be extended to other mathematical structures such as tensors, matrices, and even abstract vector spaces. The key is to define the necessary properties and axioms that the inner product must satisfy in order to be well-defined. Inner products on spinor space are just one example of this generalization.

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