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Physlib.Relativity.SL2C.SelfAdjoint

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definition

Complex 2×22 \times 2 matrices C2×2\mathbb{C}^{2 \times 2}

#termℂ²ˣ²

The notation C2×2\mathbb{C}^{2 \times 2} denotes the type of 2×22 \times 2 matrices whose entries are complex numbers C\mathbb{C}.

abbrev

Space of 2×22 \times 2 Hermitian matrices H2\mathbb{H}_2

#ℍ₂

The space H2\mathbb{H}_2 is defined as the set of 2×22 \times 2 complex matrices that are self-adjoint (Hermitian). That is, H2={M∈C2×2∣M=M†}\mathbb{H}_2 = \{ M \in \mathbb{C}^{2 \times 2} \mid M = M^\dagger \}, where M†M^\dagger denotes the conjugate transpose.

definition

The R\mathbb{R}-linear map A↦MAM†A \mapsto M A M^\dagger on H2\mathbb{H}_2

#toSelfAdjointMap'

Given a complex 2×22 \times 2 matrix M∈C2×2M \in \mathbb{C}^{2 \times 2}, this defines the R\mathbb{R}-linear map from the space of 2×22 \times 2 Hermitian matrices H2\mathbb{H}_2 to itself that sends a matrix AA to MAM†M A M^\dagger, where M†M^\dagger denotes the conjugate transpose of MM. This map is a generalization of the Lorentz transformation on Hermitian matrices where MM is not required to be in SL(2,C)SL(2, \mathbb{C}).

theorem

The determinant of A↦MAM†A \mapsto M A M^\dagger is 1 for upper triangular M∈SL(2,C)M \in SL(2, \mathbb{C})

#toSelfAdjointMap_det_one'

Let MM be a 2×22 \times 2 complex matrix that is upper triangular and has determinant det⁡M=1\det M = 1. Let H2\mathbb{H}_2 be the real vector space of 2×22 \times 2 Hermitian matrices, defined as H2={A∈C2×2∣A=A†}\mathbb{H}_2 = \{ A \in \mathbb{C}^{2 \times 2} \mid A = A^\dagger \}, where A†A^\dagger denotes the conjugate transpose. Consider the R\mathbb{R}-linear map ΦM:H2→H2\Phi_M: \mathbb{H}_2 \to \mathbb{H}_2 given by ΦM(A)=MAM†\Phi_M(A) = M A M^\dagger. Then, the determinant of the linear operator ΦM\Phi_M is equal to 11.

definition

R\mathbb{R}-linear equivalence A↦MAM†A \mapsto M A M^\dagger on H2\mathbb{H}_2

#toSelfAdjointEquiv

Given an invertible 2×22 \times 2 complex matrix M∈GL(2,C)M \in \text{GL}(2, \mathbb{C}), this defines the R\mathbb{R}-linear equivalence from the space of 2×22 \times 2 Hermitian matrices H2\mathbb{H}_2 to itself. The linear map is defined by A↦MAM†A \mapsto M A M^\dagger, and its inverse is the map A↦M−1A(M−1)†A \mapsto M^{-1} A (M^{-1})^\dagger, where M†M^\dagger denotes the conjugate transpose of MM.

theorem

ΦMN=ΦM∘ΦN\Phi_{MN} = \Phi_M \circ \Phi_N for the map ΦM(A)=MAM†\Phi_M(A) = MAM^\dagger

#toSelfAdjointMap_mul

For any 2×22 \times 2 complex matrices MM and NN, let ΦX:H2→H2\Phi_X: \mathbb{H}_2 \to \mathbb{H}_2 be the R\mathbb{R}-linear map on the space of 2×22 \times 2 Hermitian matrices defined by ΦX(A)=XAX†\Phi_X(A) = XAX^\dagger, where X†X^\dagger denotes the conjugate transpose of XX. Then the map associated with the product MNMN satisfies ΦMN=ΦM∘ΦN\Phi_{MN} = \Phi_M \circ \Phi_N.

theorem

det⁡(ΦMNM−1)=det⁡(ΦN)\det(\Phi_{MNM^{-1}}) = \det(\Phi_N) for the map ΦX(A)=XAX†\Phi_X(A) = XAX^\dagger

#toSelfAdjointMap_similar_det

Let H2\mathbb{H}_2 be the real vector space of 2×22 \times 2 Hermitian matrices. For any complex 2×22 \times 2 matrix XX, let ΦX:H2→H2\Phi_X: \mathbb{H}_2 \to \mathbb{H}_2 be the R\mathbb{R}-linear map defined by ΦX(A)=XAX†\Phi_X(A) = XAX^\dagger, where X†X^\dagger is the conjugate transpose of XX. For any 2×22 \times 2 complex matrices MM and NN such that MM is invertible, the determinant of the linear map ΦMNM−1\Phi_{MNM^{-1}} is equal to the determinant of the linear map ΦN\Phi_N.