Question

I'm exploring the spinorial representation of the Spinc

(3,1) group, especially in the context of metric preservation in general relativity and quantum field theory.

For the group GL+(4,R
)/SO(3,1) acting on matrices, the metric-preserving condition under Lorentz transformations is given by MTηM=g
, where M
is a Lorentz transformation and η
is the Minkowski metric.

I'm interested in finding an analogous operation for spinors in the context of GL+(4,R)
/Spinc
(3,1). Specifically, I'm looking for an operation ∘
such that:
ψ∘ηψ=g
where ψ
is a spinor and g
is the resulting metric or bilinear form. But here g
may not be a metric; what I mean is that I am happy with g
containing additional structure to overshoot to SO(3,1) group to Spinc
(3,1): fermions and charge.

My questions are:

What is the operation ∘
on spinors that would yield a metric-preserving condition analogous to MTηM=g
?
How does this operation account for the additional U(1)
phase factor introduced by the Spinc
structure?
Any insights or guidance on this matter would be greatly appreciated.
A) The conjugation of spinors by the Lorentz transformation matrix.
B) The action of the Spinc(3,1) group on spinors through a bispinor transformation.
C) The application of the Minkowski metric to the spinor under a spinorial transformation.
D) The tensor product of the spinors with the Minkowski metric.

Answer 1

The operation ∘ on spinors that corresponds to the metric-preserving condition MTηM=g is the Spin^c(3,1) group acting on spinors through a bispinor transformation, which includes an additional U(1) phase factor for charge.

Step-by-step explanation:

The operation ∘ on spinors that would yield a metric-preserving condition analogous to MTηM=g for Lorentz transformations is the action of the Spin^c(3,1) group on spinors through a bispinor transformation. In the context of general relativity and quantum field theory, spinors provide a way to represent particles with intrinsic spin, such as electrons. These spinors transform under the Spin^c(3,1) group, which is a double cover of the Lorentz group that also includes an additional U(1) phase factor to account for the electron charge.

The U(1) phase factor is part of the fiber bundle structure where the spinor fields live. In quantum field theory, this phase factor is related to the gauge theory of electromagnetism and represents the electromagnetic interactions of charged fermions. Therefore, the operation on spinors reflects both the geometrical aspects of spacetime symmetries and the additional gauge freedom associated with the phase of a charged particle.

Answering your question, the operation ∘ that substitutes MᴇM=g for spinors is likely to be option B, 'The action of the Spin^c(3,1) group on spinors through a bispinor transformation', which properly incorporates both Lorentz and U(1) symmetries pertinent to the Spin^c(3,1) structure, thus preserving the metric while incorporating additional fermionic charge structure.