Spin Entropy and the Lorentz Group in Biquaternionic Form

Introduction

The entropy of a spin state measures the observer's ignorance about the spin, and in a relativistic theory the natural question is how that ignorance changes from frame to frame. The framework answers it with a single algebraic object. A state of the qubit is a positive trace-one element of the Hermitian sector, $$ \tilde{\rho} = \tfrac12\bigl(e_0 + i\mathbf{r}\bigr), \qquad |\mathbf{r}|\leq 1 , $$ and its von Neumann entropy is the binary entropy of the Bloch radius, $$ S(\tilde{\rho}) = h\!\left(\frac{1+|\mathbf{r}|}{2}\right), \qquad h(p) = -p\log_2 p - (1-p)\log_2(1-p). $$ The radius is fixed by the algebra's biquaternion norm, $N(\tilde{\rho}) = \tfrac14(1-|\mathbf{r}|^2)$, so the spin entropy is a function of $N(\tilde{\rho})$ alone and is therefore an algebraic invariant: it is the same function of the biquaternion norm for every state, and it can be written $$ S(\tilde{\rho}) = h\!\left(\frac{1+\sqrt{1-4N(\tilde{\rho})}}{2}\right). $$ This is the article's central structural statement, and the Lorentz group is read through it.

Three further results follow, and they are the content of the article. First, the entropy is invariant under the compact subgroup: $SU(2)$ conjugation preserves the biquaternion norm and hence the entropy, and the entropy is in fact a complete invariant of the little-group orbit of a state, so it is the coordinate on the orbit space of the compact group. Second, the entropy is invariant under the physical action of a Lorentz transformation at sharp momentum, because that action is the unitary Wigner rotation; the entropy is not invariant under the non-unitary congruence of the algebra, but the congruence is not a state transformation, and its effect is to move a state off the trace-one slice while preserving the unnormalized biquaternion norm. Third, when the momentum is uncertain, the boost acts by a mixed-unitary channel whose effect on the entropy is an increase bounded by one bit per qubit, and the increase is governed by the Wigner angle, saturating at $h((1+\sqrt{1-v^2/c^2})/2)$ for a particle of speed $v$ transverse to the boost.

The findings are stated in advance.

1. The entropy is a function of the biquaternion norm alone. For a state, $N(\tilde{\rho}) = \tfrac14(1-|\mathbf{r}|^2)$ and $S = h((1+\sqrt{1-4N})/2)$; it vanishes on the pure states ($N = 0$) and equals one bit on the maximally mixed state ($N = \tfrac14$).
2. The compact subgroup leaves it invariant. Conjugation by any unitary element preserves $N$ and hence $S$; since $SU(2)$ acts transitively on the spheres of the Bloch ball, the entropy is constant on every $SU(2)$ orbit and is a complete invariant of that orbit.
3. The physical Lorentz action leaves it invariant at sharp momentum. The Wigner rotation is unitary, so $N(\tilde{W}\tilde{\rho}\tilde{W}^{*}) = N(\tilde{\rho})$ and the spin entropy is frame-independent for a momentum eigenstate.
4. The non-unitary congruence is not the state action. The congruence preserves the unnormalized biquaternion norm but not the trace, so after normalization the radius is scaled; the maximally mixed state is carried to $\mathbf{r} = \tanh\psi\,\hat{\mathbf{u}}$, a purer state, with the entropy $h((1+\tanh\psi)/2)$ falling towards zero. No channel can purify, which is the cleanest demonstration that the congruence is a map on the algebra and not on the state space.
5. Momentum uncertainty makes the entropy frame-dependent. The boost's channel is unital, so the maximally mixed state is fixed, but a non-maximally-mixed state is dephased by the Wigner angle and its entropy rises; for a transverse spin the entropy is $h((1+\cos\alpha)/2)$ with $\alpha$ the Wigner angle.
6. The entropy labels the little-group orbit. The quotient of the Bloch ball by $SU(2)$ is the interval of radii, and the entropy is a bijection from that interval onto $[0,1]$ bit; the Lorentz group acts on the orbit label only through the momentum, so the entropy of a sharp-momentum state is a genuine little-group invariant.

The notation is that of the foundational articles: $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$, quaternion units $e_0 = 1, e_1, e_2, e_3$ with $e_k^2 = -e_0$, central scalar $i$, trace $\mathrm{Tr}(\tilde{Q}) = 2\,\mathrm{Sc}(\tilde{Q})$, biquaternion norm $N(\tilde{Q}) = \tilde{Q}\tilde{Q}^{\natural}$, Hermitian sector $\mathbb{M}_+$, anti-Hermitian sector $\mathbb{M}_-$, and the matrix model $\Phi$. The state is $\tilde{\rho} = \tfrac12(e_0+i\mathbf{r})$; the spin operators are $\tilde{S}_k = \tfrac{\hbar}{2}ie_k$; the Wigner rotation is $\tilde{W}(\tilde{\Lambda},\tilde{U}) = \tilde{\Lambda}_{\tilde{U}'}\tilde{\Lambda}\tilde{\Lambda}_{\tilde{U}}^{-1}\in SU(2)$.

The companion articles supply the pieces: - Companion article The Relativistic Qubit in Biquaternionic Form, for the state space, the Bloch ball, and the little group. - Companion article Frame-Dependent Entanglement and Relativistic Quantum Information in Biquaternionic Form, for the channel induced by a boost on a momentum superposition. - Companion article The Wigner Rotation and the Information Content of a Boost in Biquaternionic Form, for the Wigner angle, its momentum dependence, and its holonomy. - Companion article The Bloch Ball as the Trace-One Slice of the Future Light Cone, for the geometry of the state space. - Companion article Quantum Mechanics in Biquaternionic Form, for the idempotents, the Born pairing, and the conjugation action. - Companion article The Hermitian Subspace M+ as the Informational Sector, for the trace pairing and the Hermitian sector. - Companion article The Lorentz Group in Biquaternionic Form — Structure and Representations, for the little group and the Casimir invariants.

The Spin Entropy of a State

Definition and the biquaternion-norm expression

A state of the qubit is a positive element of $\mathbb{M}_+$ of unit trace, $$ \tilde{\rho} = \tfrac12\bigl(e_0 + i\mathbf{r}\bigr), \qquad \mathbf{r}\in\mathbb{R}^3, \qquad |\mathbf{r}|\leq 1 , $$ its eigenvalues are $\tfrac12(1\pm|\mathbf{r}|)$, and its von Neumann entropy is $$ S(\tilde{\rho}) = -\sum_i\lambda_i\log_2\lambda_i = h\!\left(\frac{1+|\mathbf{r}|}{2}\right). $$ The entropy is zero on the pure states $|\mathbf{r}| = 1$, maximal and equal to $\log_2 2 = 1$ bit on the maximally mixed state $\mathbf{r} = 0$, and monotonically decreasing in $|\mathbf{r}|$. A short table of the function is

in bits.

The biquaternion norm evaluates on a state to $$ N(\tilde{\rho}) = \tilde{\rho}\tilde{\rho}^{\natural} = \tfrac14\bigl(1-|\mathbf{r}|^2\bigr)e_0 , $$ because the scalar and vector parts contribute $\tfrac14$ and $\tfrac14\sum_k(ir_k)^2 = -\tfrac14|\mathbf{r}|^2$ respectively. Hence $|\mathbf{r}| = \sqrt{1-4N(\tilde{\rho})}$ and $$ \boxed{\,S(\tilde{\rho}) = h\!\left(\frac{1+\sqrt{1-4N(\tilde{\rho})}}{2}\right)\,} $$ The entropy is a function of the single algebraic invariant $N(\tilde{\rho})$, which ranges over $[0,\tfrac14]$ for states. This is the sense in which the spin entropy is intrinsic to the algebra rather than to a chosen representation: it is computed from the biquaternion norm and the binary entropy, with no reference to a basis, a Hamiltonian, or a frame.

The Born weights and the entropy

The entropy can equally be read off from the Born pairing, which is the framework's primitive. For the two idempotents along the Bloch direction, $$ \tilde\Pi_\pm = \tfrac12\bigl(e_0 \pm i\mathbf{r}/|\mathbf{r}|\bigr), \qquad \tilde\Pi_\pm^2 = \tilde\Pi_\pm,\quad \tilde\Pi_+\tilde\Pi_- = 0,\quad \tilde\Pi_+ + \tilde\Pi_- = e_0 , $$ the Born weights of the state are the trace pairings $$ w_\pm = \frac{\mathrm{Tr}(\tilde{\rho}\tilde\Pi_\pm)}{\mathrm{Tr}(\tilde\Pi_\pm)} = \frac{1\pm|\mathbf{r}|}{2}, \qquad w_+ + w_- = 1 , $$ and the von Neumann entropy is the Shannon entropy of these two weights, $$ S(\tilde{\rho}) = -w_+\log_2 w_+ - w_-\log_2 w_- . $$ The entropy is therefore the information content of the framework's own probability rule, evaluated on the two complementary outcomes that a spin measurement along the Bloch direction can produce; the biquaternion norm and the Born pairing give the same number, as they must, because both are built from the trace and the involution.

The entropy is a complete little-group invariant

Let $\tilde{U}\in SU(2)$ be a unitary element and let $\tilde{\rho}$ be a state. The biquaternion norm is multiplicative, $$ N(\tilde{Q}_1\tilde{Q}_2) = N(\tilde{Q}_1)N(\tilde{Q}_2), $$ and $N(\tilde{U}) = N(\tilde{U}^{*}) = 1$, so $$ N(\tilde{U}\tilde{\rho}\tilde{U}^{*}) = N(\tilde{\rho}) , \qquad\text{hence}\qquad S(\tilde{U}\tilde{\rho}\tilde{U}^{*}) = S(\tilde{\rho}) . $$ The entropy is invariant under the compact subgroup — it has been verified for two thousand random rotations and random states — and it is not merely invariant: it is a complete invariant of the orbit. The action of $SU(2)$ on the Bloch ball is the rotation action, whose orbits are the spheres $|\mathbf{r}| = \text{const}$; the quotient of the state space by the group is therefore the interval of radii $[0,1]$, and the entropy is a continuous bijection from that interval onto the interval $[0,1]$ bit, $$ \text{state space}/SU(2) = [0,1]\ \xrightarrow[\ \sim\ ]{\ S\ }\ [0,1]\ \text{bit} . $$ Two states of the qubit are related by a rotation if and only if they have the same entropy. The entropy is thus the coordinate on the orbit space of the little group, and this is the group-theoretic content of the statement that "the spin is the only internal degree of freedom of a massive particle": the internal state is exhausted, up to a rotation, by its entropy.

The Lorentz Action on the Spin Entropy

The physical action at sharp momentum

The physical action of a Lorentz transformation on a spin state at definite four-velocity is the Wigner rotation, $$ \tilde{\rho} \ \longmapsto\ \tilde{W}\tilde{\rho}\tilde{W}^{*}, \qquad \tilde{W} = \tilde{W}(\tilde{\Lambda},\tilde{U})\in SU(2) . $$ Since $\tilde{W}$ is unitary, the previous computation applies verbatim, $$ N(\tilde{W}\tilde{\rho}\tilde{W}^{*}) = N(\tilde{\rho}), \qquad S(\tilde{W}\tilde{\rho}\tilde{W}^{*}) = S(\tilde{\rho}) , $$ and the spin entropy of a momentum eigenstate is a Lorentz invariant. This is the entropy form of the sharp-momentum theorem of the relativistic-qubit article, and it is exact: a frame change rotates the spin and leaves its entropy alone, because a rotation moves a state along its $SU(2)$ orbit and the entropy is constant on the orbit.

The non-unitary congruence is not a state transformation

The congruence of the algebra, $$ \tilde{\rho}\ \longmapsto\ \tilde{\Lambda}\tilde{\rho}\tilde{\Lambda}^{*} , $$ preserves the unnormalized biquaternion norm, again by multiplicativity and $N(\tilde{\Lambda}) = 1$, $$ N(\tilde{\Lambda}\tilde{\rho}\tilde{\Lambda}^{*}) = N(\tilde{\rho}) , $$ but it does not preserve the trace: $\mathrm{Tr}(\tilde{\Lambda}\tilde{\rho}\tilde{\Lambda}^{*}) = \mathrm{Tr}(\tilde{\rho}\tilde{\Lambda}^{*}\tilde{\Lambda}) = 2\,\mathrm{Sc}(\tilde{\rho}\tilde{\Lambda}^{*}\tilde{\Lambda}) \neq 1$. The physical state is the normalized element, and its biquaternion norm acquires a factor, $$ N\!\left(\frac{\tilde{\Lambda}\tilde{\rho}\tilde{\Lambda}^{*}}{\mathrm{Tr}(\tilde{\Lambda}\tilde{\rho}\tilde{\Lambda}^{*})}\right) = \frac{N(\tilde{\rho})}{\mathrm{Tr}(\tilde{\Lambda}\tilde{\rho}\tilde{\Lambda}^{*})^2} . $$ For the maximally mixed state the trace is $\cosh\psi$, and the normalized element has $$ N = \frac{1/4}{\cosh^2\psi}, \qquad |\mathbf{r}| = \sqrt{1-4N} = \tanh\psi , $$ so the congruence carries the maximally mixed state to the Bloch vector $$ \mathbf{r} = \tanh\psi\,\hat{\mathbf{u}} , $$ a state closer to the pure boundary, with entropy $h((1+\tanh\psi)/2)$ decreasing as the rapidity grows:

A genuine quantum channel is linear and trace-preserving and cannot purify a maximally mixed state; the congruence does, and the demonstration is decisive. The congruence is a map on the algebra — it is the correct action on the spinor ray and on the unnormalized projector — but it is not the transformation of the physical spin state, and the entropy is the quantity that exposes the difference most sharply.

The channel at uncertain momentum

When the momentum is uncertain, the boost acts branch by branch on the spin and the physical spin state is the output of the mixed-unitary channel $$ \tilde{\rho}\ \longmapsto\ \sum_i q_i\,\tilde{W}_i\tilde{\rho}\tilde{W}_i^{*}, \qquad \tilde{W}_i = \tilde{W}(\tilde{\Lambda},\tilde{U}_i) . $$ The channel is unital — the identity is fixed, because every $\tilde{W}_i$ is unitary — so the maximally mixed state is invariant and the entropy of a state is unchanged only if the state is a fixed point of every branch rotation. In the symmetric two-branch configuration the channel is the dephasing about the Wigner axis, the transverse Bloch components are damped by $\cos\alpha$, and the entropy becomes $h((1+\cos\alpha)/2)$ for a purely transverse initial state. For a particle of speed $0.5c$ at transverse momentum,

The entropy rises with the rapidity and saturates, in the large-rapidity limit, at $$ h\!\left(\frac{1+\sqrt{1-v^2/c^2}}{2}\right) , $$ which for $v = 0.5c$ is $h((1+0.866025)/2) = h(0.933013) = 0.354578$. Only in the ultrarelativistic limit $v\to c$ does the saturating entropy reach one bit. The entropy is therefore increased by a boost precisely when the momentum is uncertain, and the amount is a function of the Wigner angle, which is a function of the rapidity and of the angle between the boost and the momentum.

The general dephasing formula

For an arbitrary initial Bloch vector and a dephasing about the Wigner axis $\hat{\mathbf{n}}$ with damping factor $\cos\alpha$, the channel acts on the Bloch vector by $$ \mathbf{r}\ \longmapsto\ \mathbf{r}' = (\mathbf{r}\cdot\hat{\mathbf{n}})\,\hat{\mathbf{n}} + \cos\alpha\bigl(\mathbf{r} - (\mathbf{r}\cdot\hat{\mathbf{n}})\hat{\mathbf{n}}\bigr), $$ that is, it leaves the component along the axis untouched and shrinks the transverse component by $\cos\alpha$. Hence $$ |\mathbf{r}'|^2 = (\mathbf{r}\cdot\hat{\mathbf{n}})^2 + \cos^2\alpha\bigl(|\mathbf{r}|^2 - (\mathbf{r}\cdot\hat{\mathbf{n}})^2\bigr), \qquad S' = h\!\left(\frac{1+|\mathbf{r}'|}{2}\right), $$ and the entropy is unchanged only for $\mathbf{r}\parallel\hat{\mathbf{n}}$ or $\alpha = 0$. Some values for the Wigner angle of a rapidity-$1$ boost at transverse momentum, $\alpha = 14.11732^\circ$ and $\cos\alpha = 0.969798$, are

In every case the entropy increases, the increase is larger for a purer state and for transverse orientation, and the maximally mixed state $|\mathbf{r}| = 0$ is fixed for every $\alpha$. The formula also shows the sense in which the boost's effect on a spin is a partial dephasing: the component along the Wigner axis is the decoherence-free direction of the channel, and it is only this component that survives the momentum average without loss of purity.

Spin Entropy and the Little-Group Representation

Massive particles

For a massive particle the four-velocity is timelike and the little group is $SU(2)$; its irreducible representations are the spin representations, labelled by $s = 0,\tfrac12,1,\dots$, and the one-particle Hilbert space at fixed four-velocity is the carrier of the spin-$s$ representation. The entropy of a spin-$s$ state is a function on that carrier, invariant under the little group, and the orbit-space statement of the qubit case generalizes: the entropy labels the orbit of the state under the compact subgroup.

Two group-theoretic facts organise the reading. First, the mass and the spin are the Casimir labels of the Poincaré group, $$ P^2 = m^2c^2, \qquad -W^2 = m^2c^2\,s(s+1), $$ with $W^\mu$ the Pauli–Lubanski vector; they label the irreducible representation, not the state. Second, the entropy is not a Casimir; it is a function on the representation space, constant on the little-group orbits. The pair $(m,s)$ says which representation a particle belongs to; the entropy says which orbit within it a given state occupies. The Lorentz group maps the representation to itself and the orbit to an orbit, and at sharp momentum it maps each orbit to itself, because the map is the unitary little-group action. This is the precise group-theoretic sense in which spin entropy is a frame-independent property of a momentum eigenstate.

Massless particles and helicity

For a massless particle the four-momentum is null and the little group is the two-dimensional Euclidean group $E(2)$, whose finite-dimensional unitary representations are one-dimensional and labelled by the helicity. A helicity eigenstate has no internal entropy at all: the internal state space is one-dimensional, the entropy is zero, and a Lorentz transformation can only multiply the state by a phase, leaving the entropy zero. The polarization entropy of a massless particle is therefore identically zero for a helicity eigenstate, and any positive entropy in a massless state is entropy of a superposition of helicities, which the little group does not mix. This is the sharpest contrast with the massive case: the massive qubit has a one-parameter family of entropies, the massless helicity state has none.

The entropy of the compact subgroup

The two cases can be stated together. The little group is a subgroup of the unit-norm group, and the entropy of a state is a function of the biquaternion norm, which is invariant under the unitary part of the group and only under it. The unitary part is $SU(2)$ for a massive particle and a phase group for a massless one; correspondingly, the spin entropy is a non-trivial function on the state space in the massive case and a trivial one in the massless case. The entropy therefore measures the size of the compact part of the little group available to the state, which is the representation-theoretic content of the statement that spin entropy is a relativistic quantity: it exists because the massive little group is non-abelian.

The Two Regimes of Frame Dependence

Sharp momentum. The spin entropy is a Lorentz invariant. The physical transformation is the unitary Wigner rotation, it preserves the biquaternion norm, and the entropy is a function of the biquaternion norm; the result is exact and holds for every rapidity and every boost direction.

Uncertain momentum. The spin entropy is frame-dependent, increasing under a boost by the dephasing produced by the branch-dependent Wigner rotations. The increase is bounded by one bit for a qubit, saturated in the ultrarelativistic limit, and equal to $h((1+\cos\alpha)/2)$ for a transverse spin in the symmetric two-branch configuration. It is invisible in the maximally mixed state, which is a fixed point of the unital channel.

The contrast with the congruence. The non-unitary congruence produces the opposite behaviour: it decreases the entropy of the maximally mixed state, from one bit towards zero, by scaling the trace and renormalizing. The contrast is the operational test that separates the two actions, and it can be stated in one sentence: a physical transformation of a state cannot create purity, and the congruence does; the physical transformation at sharp momentum is the unitary Wigner rotation, and at uncertain momentum it is the unital channel built from the Wigner rotations of the branches.

What is invariant. The entropy is invariant under $SU(2)$, under the unitary little group of the massive case, under the phase little group of the massless case, and along every orbit of these groups; it is invariant under the classical Lorentz action on the material sector (which does not touch the spin at all); and it is invariant under the physical action of the Lorentz group at sharp momentum. It is not invariant under the non-unitary congruence, which is not a state transformation, and it is not invariant under the physical action at uncertain momentum, which is a channel.

The Entropy and the Biquaternion Norm

The single most compact statement of the article is that the spin entropy is a function of the algebra's biquaternion norm. For a state, $$ N(\tilde{\rho}) = \tfrac14\bigl(1-|\mathbf{r}|^2\bigr), \qquad S(\tilde{\rho}) = h\!\left(\frac{1+\sqrt{1-4N(\tilde{\rho})}}{2}\right), $$ so the entropy is not an independent quantity attached to the state from outside; it is the binary entropy of the algebraic invariant $N$, read on the trace-one slice. The Lorentz group is a group of norm-preserving congruences on the algebra, and its action on the entropy is therefore controlled entirely by what it does to the trace and to the biquaternion norm:

• the unitary subgroup preserves the trace as well, so the entropy is invariant;
• the unit-norm, non-unitary elements preserve the biquaternion norm but not the trace, so the entropy of the normalized element changes, and the direction of the change — purity increase — shows that the congruence is not a state transformation;
• the momentum-averaged channel replaces the single congruence by an average of unitary ones, restoring trace preservation and therefore making the change of entropy a genuine change of state, driven by the Wigner angles of the branches.

This is the whole content of the Lorentz group's action on spin entropy, and it is a statement about the biquaternion norm and the trace.

Summary

The spin state of the qubit is $\tilde{\rho} = \tfrac12(e_0+i\mathbf{r})$ with $|\mathbf{r}|\leq1$, and its von Neumann entropy is $$ S(\tilde{\rho}) = h\!\left(\frac{1+|\mathbf{r}|}{2}\right) = h\!\left(\frac{1+\sqrt{1-4N(\tilde{\rho})}}{2}\right), \qquad h(p) = -p\log_2 p - (1-p)\log_2(1-p), $$ a function of the algebra's biquaternion norm alone. It vanishes on the pure states, where $N = 0$, and equals one bit on the maximally mixed state, where $N = \tfrac14$.

The entropy is invariant under the compact subgroup, because $N(\tilde{U}\tilde{\rho}\tilde{U}^{*}) = N(\tilde{\rho})$ for every unitary $\tilde{U}$, and it is a complete invariant of the $SU(2)$ orbit: the quotient of the state space by the little group is the interval of radii, and the entropy is a bijection from it onto $[0,1]$ bit. It is invariant under the physical Lorentz action at sharp momentum, which is the unitary Wigner rotation $\tilde{\rho}\mapsto\tilde{W}\tilde{\rho}\tilde{W}^{*}$, and it is not invariant under the non-unitary congruence $\tilde{\rho}\mapsto\tilde{\Lambda}\tilde{\rho}\tilde{\Lambda}^{*}$, which preserves the unnormalized biquaternion norm but not the trace and carries the maximally mixed state to $\mathbf{r} = \tanh\psi\,\hat{\mathbf{u}}$ with entropy $h((1+\tanh\psi)/2)$ — a purification, and therefore a proof that the congruence is not a state transformation.

At uncertain momentum the boost acts by the unital mixed-unitary channel $\tilde{\rho}\mapsto\sum_iq_i\tilde{W}_i\tilde{\rho}\tilde{W}_i^{*}$; the maximally mixed state is a fixed point, and a transverse spin is dephased by the Wigner angle, its entropy becoming $h((1+\cos\alpha)/2)$. For a particle of speed $0.5c$ at transverse momentum the entropy is $0.039902$, $0.112969$, $0.242196$, $0.309540$, and $0.348220$ at rapidities $0.5$, $1$, $2$, $3$, and $5$, saturating at $h((1+\sqrt{1-v^2/c^2})/2) = 0.354578$. The increase is bounded by one bit and saturates only in the ultrarelativistic limit.

For a massive particle the little group is $SU(2)$, the Casimir labels are $(m,s)$, and the entropy is a function on the representation space, constant on the little-group orbits: the pair $(m,s)$ names the representation and the entropy names the orbit within it. For a massless particle the little group is $E(2)$, the internal space of a helicity eigenstate is one-dimensional, and the spin entropy is identically zero. The entropy is a relativistic quantity because the massive little group is non-abelian, and it is frame-independent exactly when the momentum is sharp.

Summary of Notation

Further Reading

• E. P. Wigner, "On unitary representations of the inhomogeneous Lorentz group," Annals of Mathematics 40 (1939) 149–204, for the little groups and the Casimir classification.
• Eugene P. Wigner, "Relativistic invariance and quantum phenomena," Reviews of Modern Physics 29 (1957) 255–268, for the physical reading of the little group and its representations.
• S. Weinberg, The Quantum Theory of Fields, Vol. 1 (Cambridge, 1995), for the one-particle representations, the Pauli–Lubanski vector, and the little-group construction.
• Asher Peres, Petra F. Scudo, and Daniel R. Terno, "Quantum entropy and special relativity," Physical Review Letters 88 (2002) 230402, for the frame dependence of the reduced spin entropy.
• Asher Peres and Daniel R. Terno, "Quantum information and relativity theory," Reviews of Modern Physics 76 (2004) 93–123, for the relativistic quantum-information setting.
• Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information (Cambridge, 2000), for the von Neumann entropy, the Bloch ball, and the qubit state space.
• Ingemar Bengtsson and Karol Życzkowski, Geometry of Quantum States (Cambridge, 2006), for the geometry of the state space, its symmetry groups, and the orbit structure.
• Chris Doran and Anthony Lasenby, Geometric Algebra for Physicists (Cambridge, 2003), for the rotor description of the Lorentz group and its compact subgroup.
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge, 2001), for the structure of the spinor representation and its invariant forms.