spacetime-momentum-spinor-representation

Prop Generally, for , are not equivalent in the sense of changing coordinates: There does not exist ,

Proof

Eigenvalues remain unchanged under coordinate transformation

Generally, have different eigenvalues

Example

Prop are equivalent, are equivalent

use

Its complex conjugate [conjugate-representation]

The above only works for

Second-order tensor, with one undergoing complex conjugation

can be decompose to

[Hermitian-tensor]

[anti-Hermitian-tensor]

The conjugate modification in the other direction is similar

[Hermitian-tensor-induced-linear-map] The induced action of on :=

[matrix-description-of-Hermitian-tensor]

Use tensor base

Corresponding to the matrix representation

Or written in Dirac notation

The version without complex conjugation

notation-overload: The space of matrix representation is also denoted as

Hermitian matrix

anti-Hermitian matrix

For , since , the dimension of anti-Hermitian is higher than Hermitian

[Hermitian-tensor-induced-linear-map-matrix] The matrix representation of

preserves the decomposition into Hermitian and anti-Hermitian

For general , there is also

The “matrix” representation of needs to be handled separately, the composition of cannot be represented as usual matrix multiplication. It can still make well-defined

[spacetime-momentum-spinor-representation]

( represents “momentum” or “velocity” or tangent vector)

Bijection

metric

let and , action

Because multiplication is non-commutative, the general definition of for is problematic. However, the definition of does not require general multiplicative commutativity. At this time, is defined as . This is not a good notation because it may not be generalized to

are also the spinor lifts of . Similarly, is also the spinor lift of

Example [Pauli-matrix] alias [sigma-matrix]

for

time-like
light-like
space-like

(when generalized to , it corresponds to all imaginary elements)

is orthonormal base

Question What is the cognitive motivation for these constructions of ?

acts on lifts to act on

action, denoted as
action, denoted as

[square-root-of-Lorentz-group]

act on is some kind of “square” of , i.e. or represents the “real part” or “symmetric part”

Thus are some kind of “square root” of act on

[square-root-of-spacetime-metric-1] (inspired by ref-14, ch.11)

. Note that it is not alternating for

metric with is some kind of “square” of , i.e.

quadratic-form is

cf. Pauli-matrix

The calculation shows that is correct for . For , use sum

orthogonal of sigma matrix can also be obtained through calculation, thus

Thus is some kind of “square root” of metric

Question Still no direct intuitive calculation equation …

[spacetime-momentum-aciton-spinor-representation]

let .

where is Lorentz-group-spinor-representation

is spacetime-momentum-spinor-representation

Then there is a homomorphism

Proof Use the correspondence of 3 rotations, 3 boosts

[spinor-representation-adjoint]

Proof

use 3 boost, 3 rotation

use

Prop Use spacetime-momentum-spinor-representation for , + projection gives projective-lightcone

Therefore the following are equivalent

act on via
act on via

Proof

with (requires associative law of multiplication?)

Given

in ,

let

Also need to calculate

In order to get , compare norm, phase

norm

phase

so let with

Generally . Compare to get

[parity]

parity corresponds to vs representation, or vs , cf. conjugate-representation

let .

parity corresponds to space inversion

corresponds to time inversion

parity corresponds to trace or determinant reversal

[determinant-reversal]

let

determinant reversal with

[trace-reversal] := . or .

==> determinant reversal is the same as trace reversal

[square-root-of-spacetime-metric-2] a “square root” of metric

let .

give

Also have

for Pauli-matrix

or
, for (because parity is spatial inversion)

A better explanation of this “square root”?

Direct matrix multiplication without parity will give the square root of the metric, with ,

[square-root-of-Lorentz-Lie-algebra] “square root” of spacetime Lie-algebra

where is Lorentz-Lie-algebra

Proof

is δ rotation in where is any cyclic
where

Question A better explanation? Representation?

[property-of-parity]

i.e. parity preserve Hermitian

[parity-Euclidean-invariant] parity commutes with spatial action . In , it manifests as and commutes with . let

Generally, they do not commute, for example, certainly does not commute with the time-changing part in

let

[parity-reverse-boost] The effect of parity on the Lie-algebra is that it does not change δ rotation, but multiplies δ boost by

[Euclidean-spinor]

replace lightcone with just sphere acted by and

replace with , with

use trace-free Hermitian

note-math