2026-10-26
Arbitrary Spin in Splitting Operators: Rotations and Equations of Motion
Rotating a quantum state can produce a result unfamiliar in ordinary geometry: after one full turn, the amplitude changes sign, and after two full turns it returns to its original value. In the previous article, we obtained this behavior for spin 1/2 using channel selection and exchange. We will now use the same operations to construct spins 1, 3/2, and 2, derive the general rule, and show how rotations can be written as products of J operators. We will then examine the transition to relativistic equations for free fields.
This article continues The Dirac Equation in Splitting Operators. The notation and physical postulates are introduced in the Wave Electricity Concept. Spacetime is taken as given throughout this article; the origins of space, mass, and interactions are not derived here.
1. A single binary spin factor
We begin with two orthonormal channels, \(|+\rangle\) and \(|-\rangle\). The idempotents select them, the hyperbolic unit assigns them opposite signs, and the operator \(S\) exchanges the channels:
\[\tag{1} \ep^2=\ep,\quad \em^2=\em,\quad \ep\em=0,\quad \ep+\em=I,\qquad \j=\ep-\em,\qquad S|+\rangle=|-\rangle,\quad S|-\rangle=|+\rangle. \] Here \(\ep=|+\rangle\langle+|\) and \(\em=|-\rangle\langle-|\), while \(S\) is extended to all states by complex linearity. Thus \(\ep,\em\) are orthogonal Hermitian projectors, and \(\j\) and \(S\) are Hermitian operators with respect to the chosen inner product.
The definitions imply \(\j^2=S^2=I\) and \(S\j=-\j S\). As shown in the previous article, the three Hermitian operators
\[\tag{2} q_1=S,\qquad q_2=iS\j,\qquad q_3=\j \] satisfy the relations
\[\tag{3} [q_i,q_j]=2i\varepsilon_{ijk}q_k,\qquad \{q_i,q_j\}=2\delta_{ij}I. \] The internal rotation generators \(\widehat s_i=\hbar q_i/2\) therefore give
\[\tag{4} \widehat{\mathbf s}^{\,2}=\frac34\hbar^2I,\qquad \widehat s_z|\pm\rangle=\pm\frac{\hbar}{2}|\pm\rangle. \] This is a representation with spin \(s=1/2\). Here and below, \(\hbar\) retains the dimensions of action, angles are measured in radians, and \(s\) and the projection quantum numbers \(\mu\) are dimensionless. The physical angular momentum projection is \(\hbar\mu\). These internal generators do not include the orbital contribution to the rotation of the spatial wavefunction.
A rotation about the axis specified by the unit vector \(\mathbf n\) takes the form
\[\tag{5} q_{\mathbf n}=\mathbf n\cdot\mathbf q,\qquad P_{\mathbf n,\pm}=\frac{I\pm q_{\mathbf n}}2,\qquad R_{\mathbf n}^{(1/2)}(\theta) =P_{\mathbf n,+}e^{-i\theta/2}+P_{\mathbf n,-}e^{i\theta/2}. \] We introduce a phase operator of the same form as \(J\), using projectors onto the spin channels of the chosen axis:
\[\tag{6} J_{\mathbf n}(a_{\mathbf n},b_{\mathbf n}) =P_{\mathbf n,+}e^{i\pi b_{\mathbf n}}+ P_{\mathbf n,-}e^{i\pi a_{\mathbf n}},\qquad R_{\mathbf n}^{(1/2)}(\theta) =J_{\mathbf n}\left(\frac{\theta}{2\pi},-\frac{\theta}{2\pi}\right). \] The parameters \(a_{\mathbf n},b_{\mathbf n}\) here specify rotation phases. They do not replace the internal parameter \(a\) or the external-motion parameter \(b\) of the original physical operator \(J(a,b)\).
2. Independent factors and total spin
To generalize the construction, take \(N\) independent binary spin factors. Their state space is the tensor product \((\mathbb C^2)^{\otimes N}\). The index \(r\) labels a factor, not a spatial coordinate. Operators acting on different factors commute:
\[\tag{7} [q_i^{(r)},q_j^{(t)}]=0\quad(r\ne t),\qquad \widehat T_i=\frac{\hbar}{2}\sum_{r=1}^{N}q_i^{(r)}. \] The rotation algebra follows directly:
\[\tag{8} [\widehat T_i,\widehat T_j] =\frac{\hbar^2}{4}\sum_r[q_i^{(r)},q_j^{(r)}] =i\hbar\varepsilon_{ijk}\widehat T_k. \] This algebra alone does not imply that the entire space has a single definite spin. In general, the space decomposes into several independent sectors. We first examine the simplest example; the standard theory of angular momentum addition is presented in [1].
The tensor factors describe the internal structure of the state space. Introducing them does not, by itself, mean that a particle consists of several physical particles. Symmetry in these indices likewise makes no claim about particle exchange or statistics.
3. Two factors: spins 1 and 0
For \(N=2\), there are four initial channels: \(|++\rangle,|+-\rangle,|-+\rangle,|--\rangle\). The three states symmetric under interchange of the factors form a triplet:
\[\tag{9} |1,1\rangle=|++\rangle,\qquad |1,0\rangle=\frac{|+-\rangle+|-+\rangle}{\sqrt2},\qquad |1,-1\rangle=|--\rangle. \] The first label in \(|s,\mu\rangle\) is the spin, and the second is its projection quantum number. For these states,
\[\tag{10} \widehat T_z|1,\mu\rangle=\hbar\mu|1,\mu\rangle,\qquad \widehat{\mathbf T}^{\,2}|1,\mu\rangle=2\hbar^2|1,\mu\rangle. \] The last equality can be checked without writing out matrices. For two factors,
\[\tag{11} \widehat{\mathbf T}^{\,2} =\frac32\hbar^2I+\frac{\hbar^2}{2} \sum_{i=1}^{3}q_i^{(1)}q_i^{(2)}. \] Denote the interchange of the two factors by \(\Pi_{12}\), to distinguish it from channel exchange \(S\) within a single factor. Its action is defined by \(\Pi_{12}|ab\rangle=|ba\rangle\). Checking the four basis channels gives
\[\tag{12} \Pi_{12}=\frac12\left(I+\sum_iq_i^{(1)}q_i^{(2)}\right),\qquad \widehat{\mathbf T}^{\,2}=\hbar^2(I+\Pi_{12}). \] On symmetric states, \(\Pi_{12}=+I\), so the total spin is 1. The remaining combination is antisymmetric:
\[\tag{13} |0,0\rangle=\frac{|+-\rangle-|-+\rangle}{\sqrt2},\qquad \Pi_{12}|0,0\rangle=-|0,0\rangle,\qquad \widehat T_i|0,0\rangle=0. \] This is a spin-0 singlet. Thus,
\[\tag{14} \frac12\otimes\frac12=1\oplus0,\qquad 4=3+1. \] Equal numbers of plus and minus signs give zero projection, but not necessarily zero spin. Both the triplet state \(|1,0\rangle\) and the singlet state \(|0,0\rangle\) have \(\mu=0\); they differ in their transformations under rotations about other axes and in the value of \(\widehat{\mathbf T}^{\,2}\).
For a rotation about \(z\), each plus sign contributes the phase factor \(e^{-i\theta/2}\), and each minus sign contributes \(e^{i\theta/2}\). On the triplet, therefore,
\[\tag{15} R_z^{(1)}(\theta)|1,\mu\rangle=e^{-i\mu\theta}|1,\mu\rangle,\qquad R_z^{(1)}(\theta)=P_{z,1}e^{-i\theta}+P_{z,0}+P_{z,-1}e^{i\theta}. \] Every internal rotation acts as the identity on the singlet. This is a statement about spin: the spatial dependence of a scalar field still transforms when the coordinates are rotated.
4. Three factors: spin 3/2
Three binary factors give eight initial channels. The fully symmetric states form a four-dimensional subspace:
\[\tag{16} \begin{aligned} \left|\frac32,\frac32\right\rangle&=|+++\rangle,\\ \left|\frac32,\frac12\right\rangle&=\frac{|++-\rangle+|+-+\rangle+|-++\rangle}{\sqrt3},\\ \left|\frac32,-\frac12\right\rangle&=\frac{|+--\rangle+|-+-\rangle+|--+\rangle}{\sqrt3},\\ \left|\frac32,-\frac32\right\rangle&=|---\rangle. \end{aligned} \] The coefficients \(1/\sqrt3\) ensure unit norm, since the three distinct channels are orthogonal. These states have
\[\tag{17} \widehat{\mathbf T}^{\,2}=\frac{15}{4}\hbar^2I,\qquad \mu=\frac32,\frac12,-\frac12,-\frac32. \] The general derivation of this eigenvalue is given below. Here we can check the action of the raising and lowering operators. For example,
\[\tag{18} \widehat T_-|+++\rangle =\hbar\bigl(|-++\rangle+|+-+\rangle+|++-\rangle\bigr) =\sqrt3\hbar\left|\frac32,\frac12\right\rangle. \] The full eight-dimensional space decomposes as
\[\tag{19} \left(\frac12\right)^{\otimes3} =\frac32\oplus\frac12\oplus\frac12,\qquad 8=4+2+2. \] The two remaining spin-\(1/2\) sectors have mixed permutation symmetry. They do not belong to the selected fully symmetric subspace. For example, the state \((|+-+\rangle-|-++\rangle)/\sqrt2\) is obtained from the singlet of the first two factors and the third binary factor.
A rotation about \(z\) multiplies the four symmetric components by \(e^{-3i\theta/2},e^{-i\theta/2},e^{i\theta/2},e^{3i\theta/2}\). At \(\theta=2\pi\), all four factors equal \(-1\); at \(4\pi\), they equal \(+1\).
5. Four factors: spin 2
Four binary factors give sixteen initial channels. There are five fully symmetric states:
\[\tag{20} \begin{aligned} |2,2\rangle&=|++++\rangle,\\ |2,1\rangle&=\frac{|-+++\rangle+|+-++\rangle+|++-+\rangle+|+++-\rangle}{2},\\ |2,0\rangle&=\frac1{\sqrt6}\bigl(|++--\rangle+|+-+-\rangle+|+--+\rangle\\ &\qquad{}+|-++-\rangle+|-+-+\rangle+|--++\rangle\bigr),\\ |2,-1\rangle&=\frac{|+---\rangle+|-+--\rangle+|--+-\rangle+|---+\rangle}{2},\\ |2,-2\rangle&=|----\rangle. \end{aligned} \] The numbers of initial combinations are \(1,4,6,4,1\); these determine the normalization. We obtain
\[\tag{21} \widehat{\mathbf T}^{\,2}=6\hbar^2I,\qquad \widehat T_z|2,\mu\rangle=\hbar\mu|2,\mu\rangle,\qquad \mu=2,1,0,-1,-2. \] The rotation phase factors are \(e^{-2i\theta},e^{-i\theta},1,e^{i\theta},e^{2i\theta}\). All spin-2 states return to their original values after \(2\pi\). The full space also contains lower spins:
\[\tag{22} \left(\frac12\right)^{\otimes4} =2\oplus1\oplus1\oplus1\oplus0\oplus0,\qquad 16=5+3+3+3+1+1. \] 6. The general construction: s = N/2
Let \(k\) be the number of factors carrying a minus sign among the \(N\) factors. Write \(\mathcal C_{N,k}\) for the set of all sequences \(\boldsymbol\varepsilon=(\varepsilon_1,\ldots,\varepsilon_N)\) with \(\varepsilon_r=\pm1\) and exactly \(k\) negative signs. The signs \(+1,-1\) correspond to the channels \(|+\rangle,|-\rangle\), respectively. There are \(\binom Nk\) distinct sequences, and the normalized symmetric state is
\[\tag{23} \begin{aligned} |s,\mu\rangle&= \frac{1}{\sqrt{\binom Nk}} \sum_{\boldsymbol\varepsilon\in\mathcal C_{N,k}} |\varepsilon_1\ldots\varepsilon_N\rangle,\\ s&=\frac N2,\qquad \mu=\frac N2-k,\qquad k=0,\ldots,N. \end{aligned} \] No sequence is repeated in this sum. Different values of \(k=0,\ldots,N\) give orthogonal states, so the symmetric subspace has dimension \(N+1\).
To determine the total spin, introduce \(\widehat T_\pm=\widehat T_x\pm i\widehat T_y\). Their action on these states follows by counting the signs that can be flipped:
\[\tag{24} \begin{aligned} \widehat T_z|s,\mu\rangle&=\hbar\mu|s,\mu\rangle,\\ \widehat T_+|s,\mu\rangle&=\hbar\sqrt{(s-\mu)(s+\mu+1)}\,|s,\mu+1\rangle,\\ \widehat T_-|s,\mu\rangle&=\hbar\sqrt{(s+\mu)(s-\mu+1)}\,|s,\mu-1\rangle. \end{aligned} \] At the ends of the sequence, \(\widehat T_+|s,s\rangle=0\) and \(\widehat T_-|s,-s\rangle=0\): no states with projections outside the range \(-s,\ldots,s\) are introduced.
For example, raising the projection changes one of the \(k\) minus signs into a plus. Accounting for repeated contributions and the normalization of adjacent sums gives the coefficient \(\hbar\sqrt{k(N-k+1)}\), in agreement with the second line.
We use the identity
\[\tag{25} \widehat{\mathbf T}^{\,2}=\widehat T_z^{\,2} +\frac12(\widehat T_+\widehat T_-+\widehat T_-\widehat T_+). \] Substituting the preceding expressions gives the same eigenvalue on every state. Hence,
\[\tag{26} \boxed{ \mathcal H_s=\operatorname{Sym}^{2s}(\mathbb C^2),\qquad \dim\mathcal H_s=2s+1,\qquad \widehat{\mathbf T}^{\,2}=\hbar^2s(s+1)I. } \] The symmetric subspace is selected by the orthogonal projector \(\mathcal P_{\mathrm{sym}}=(1/N!)\sum_{\pi\in\mathfrak S_N}\Pi_\pi\), where \(\mathfrak S_N\) is the group of permutations of the factors, and \(\Pi_\pi\) permutes the factors according to \(\pi\). Its image is \(\mathcal H_s\), and \([\mathcal P_{\mathrm{sym}},\widehat T_i]=0\). A collective rotation therefore keeps the state within the selected spin sector. Applying the projector to an arbitrary state can reduce its norm; preparing a normalized state requires normalizing the resulting nonzero vector.
Here \(\operatorname{Sym}\) denotes the fully symmetric tensor subspace. For \(s=0\), one can directly use the one-dimensional scalar representation, corresponding to \(N=0\). The two-factor singlet is another realization of the same rotation representation; it does not imply that every scalar field must have two hidden components.
7. Arbitrary rotations through J
For a fixed axis, generators acting on different factors commute. The exponential of their sum therefore factorizes:
\[\tag{27} R_{\mathbf n}(\theta) =\exp\left(-\frac{i\theta}{2}\sum_{r=1}^{N}q_{\mathbf n}^{(r)}\right) =\prod_{r=1}^{N}\exp\left(-\frac{i\theta}{2}q_{\mathbf n}^{(r)}\right). \] This is a product of operators on different tensor factors: each operator acts on its own factor and as the identity on all others. It cannot be reduced using the composition rule \(J(a,b)J(u,v)=J(a+u,b+v)\), which applies to a single pair of projectors.
For each factor, define
\[\tag{28} P_{\mathbf n,\pm}^{(r)}=\frac{I\pm q_{\mathbf n}^{(r)}}2,\qquad J_{\mathbf n}^{(r)}(a_r,b_r) =P_{\mathbf n,+}^{(r)}e^{i\pi b_r} +P_{\mathbf n,-}^{(r)}e^{i\pi a_r}. \] Applying the same rotation to all factors preserves the symmetric subspace. Restricting to that subspace gives the desired formula:
\[\tag{29} \boxed{ R_{\mathbf n}^{(s)}(\theta)= \left. \prod_{r=1}^{2s} J_{\mathbf n}^{(r)} \left(\frac{\theta}{2\pi},-\frac{\theta}{2\pi}\right) \right|_{\mathcal H_s} = \sum_{\mu=-s}^{s}P_{\mathbf n,\mu}e^{-i\mu\theta}. } \] The projectors \(P_{\mathbf n,\mu}\) on the right select the eigenspaces of \(\widehat T_{\mathbf n}=\mathbf n\cdot\widehat{\mathbf T}\) within \(\mathcal H_s\). In the next formula, both \(\widehat T_{\mathbf n}\) and the identity \(I\) act specifically on \(\mathcal H_s\). This restriction matters: on the full tensor space, a given projection does not distinguish different spin sectors. The spectral projectors of the selected sector are explicitly
\[\tag{30} P_{\mathbf n,\mu} =\prod_{\substack{\nu=-\nu\ne\mu}}^{s} \frac{\widehat T_{\mathbf n}/\hbar-\nu I}{\mu-\nu},\qquad P_{\mathbf n,\mu}P_{\mathbf n,\nu}=\delta_{\mu\nu}P_{\mathbf n,\mu}, \qquad \sum_{\mu=-s}^{s}P_{\mathbf n,\mu}=I_{\mathcal H_s}. \] For \(s=0\), empty products are understood as the identity. For \(s=1/2\), there are two spin projectors. For higher spin, there are \(2s+1\), so a single binary \(J\) is generally insufficient to describe the entire one-parameter family of rotations. A product of binary \(J\) operators achieves this without changing the elementary operator.
For a channel with \(\varepsilon_r=\pm1\), the total phase factor is
\[\tag{31} \prod_{r=1}^{N}e^{-i\varepsilon_r\theta/2} =e^{-i\mu\theta},\qquad \mu=\frac12\sum_{r=1}^{N}\varepsilon_r. \] This gives the general rule for a full turn:
\[\tag{32} \boxed{ R_{\mathbf n}^{(s)}(2\pi)=(-1)^{2s}I,\qquad R_{\mathbf n}^{(s)}(4\pi)=I. } \] The overall sign of a state does not change probabilities in an isolated branch. A relative sign between coherent branches is observable. Neutron interferometry provides an experimental test of spinor periodicity [4]. This supports the spin-1/2 rotation law, but does not by itself establish the physical interpretation of channels in this model.
8. What is needed for an equation of motion
So far, we have constructed an internal representation of rotations. It determines spin, but does not specify mass, energy, spatial derivatives, or transformations between inertial frames. A relativistic field requires an additional construction.
In the previous article, a single four-component Dirac factor was constructed from two binary factors. To distinguish these from the numbering of spin factors, denote their internal operations by \(\j_E,S_E\) and \(\j_\sigma,S_\sigma\). The index \(E\) labels the first factor; at nonzero momentum, it is not automatically the sign of the energy. The operators are
\[\tag{33} B=\j_E,\qquad A_1=S_ES_\sigma,\qquad A_2=iS_ES_\sigma\j_\sigma,\qquad A_3=S_E\j_\sigma, \qquad \Gamma^0=B,\quad\Gamma^i=BA_i. \] They satisfy \(\{\Gamma^\lambda,\Gamma^\rho\}=2\eta^{\lambda\rho}I\), where \(\eta=\operatorname{diag}(1,-1,-1,-1)\) and \(x^0=ct\). Here the Greek indices \(\lambda,\rho\) refer to spacetime, while \(\mu\) continues to denote the spin projection quantum number.
For a free massive field of spin \(s>0\), use the standard Bargmann–Wigner construction [2]. Take \(N=2s\) Dirac indices, each ranging from 1 to 4, and a field fully symmetric in these indices:
\[\tag{34} \Psi_{\alpha_1\ldots\alpha_N}(x) =\Psi_{\alpha_{\pi(1)}\ldots\alpha_{\pi(N)}}(x) \quad\text{for any permutation }\pi. \] A Lorentz transformation acts on all indices simultaneously: \(\Psi'(x')=L^{\otimes N}\Psi(x)\). If \(L^{-1}\Gamma^\lambda L=\Lambda^\lambda{}_{\rho}\Gamma^\rho\), each equation below transforms covariantly. The same action on all factors preserves the symmetry of the field.
A Dirac equation with the same mass \(m>0\) is imposed on each index:
\[\tag{35} \boxed{ (i\hbar\Gamma_{(r)}^\lambda\partial_\lambda-mc)\Psi=0, \qquad r=1,\ldots,2s. } \] The index \((r)\) specifies the Dirac factor on which the operator acts. The same coordinate \(x\) is used throughout: this is a system of equations for a single field. In our notation, each \(\Gamma_{(r)}^\lambda\) is constructed from splittings and exchanges using the formula above. This expresses the established covariant system in the operator language of this article.
Apply the operator \(i\hbar\Gamma_{(r)}^\lambda\partial_\lambda+mc\) from the left; its mass term has the opposite sign to that in the original equation. This is an algebraic factorization, not Hermitian conjugation. Since the derivatives commute, the antisymmetric part of the product of \(\Gamma\) operators contributes nothing. We obtain the second-order consistency check:
\[\tag{36} \left(\Box+\frac{m^2c^2}{\hbar^2}\right)\Psi=0,\qquad \Box=\frac1{c^2}\partial_t^2-\nabla^2. \] The spin must also be checked. For a plane wave in the rest frame with positive energy \(mc^2\), each equation requires \(B_{(r)}\Psi=\Psi\). Two components remain in each Dirac factor. Their symmetric product is precisely \(\operatorname{Sym}^{2s}(\mathbb C^2)\), with the spin \(s\) derived above. Similarly, the negative-energy branch has \(B_{(r)}\Psi=-\Psi\). A nonzero mass allows a transformation from the rest frame to any inertial frame.
The number of covariant field components before imposing the equations should not be confused with the number of polarizations. A symmetric multispinor has \(\binom{N+3}{3}\) algebraically independent components; at fixed momentum, the equations leave \(2s+1\) polarizations on each energy branch. Interpreting these branches in terms of particles and antiparticles requires field quantization.
The formula using N binary spin factors describes rotations in spin space. The Bargmann–Wigner construction uses N four-component Dirac factors, or 2N binary factors before symmetrization and the imposition of the equations and constraints. These two counts refer to different stages of the construction.
9. Examples of free massive dynamics
Spin 0. For a scalar field \(\phi(x)\), the internal rotation is the identity, and we specify the dynamics by the Klein–Gordon equation:
\[\tag{37} \left(\Box+\frac{m^2c^2}{\hbar^2}\right)\phi=0. \] For \(N=0\), the Bargmann–Wigner system is empty and does not itself yield this equation: it must be specified separately. Zero spin also does not imply an absence of spatial motion or orbital angular momentum.
Spin 1/2. A single Dirac index gives the original equation:
\[\tag{38} (i\hbar\Gamma^\lambda\partial_\lambda-mc)\Psi=0. \] Spin 1. With two symmetric Dirac indices, the system can be expressed in the Proca tensor form for a vector field \(A^\lambda\) [3]:
\[\tag{39} F^{\lambda\rho}=\partial^\lambda A^\rho-\partial^\rho A^\lambda,\qquad \partial_\lambda F^{\lambda\rho} +\frac{m^2c^2}{\hbar^2}A^\rho=0. \] Taking the divergence of the second equation for \(m>0\) gives \(\partial_\rho A^\rho=0\). Consequently,
\[\tag{40} \left(\Box+\frac{m^2c^2}{\hbar^2}\right)A^\rho=0,\qquad \partial_\rho A^\rho=0. \] In the rest frame, the transversality condition removes the temporal component of a nonzero-frequency mode. Three independent spatial polarizations remain, corresponding to the three spin-1 states. The standard equivalent tensor form is shown here; the explicit conversion of multispinor components into \(A^\lambda,F^{\lambda\rho}\) requires a separate calculation.
Spin 3/2. Three symmetric Dirac indices can be replaced by a vector-spinor description. For a free massive particle, a useful on-shell form of the Rarita–Schwinger system is [3, 5]:
\[\tag{41} (i\hbar\Gamma^\lambda\partial_\lambda-mc)\psi_\rho=0,\qquad \Gamma^\rho\psi_\rho=0,\qquad \partial^\rho\psi_\rho=0. \] At fixed energy branch in the rest frame, the vector-spinor before the additional constraints contains a spin-\(3/2\) sector and two spin-\(1/2\) sectors: the temporal vector component gives one of them, and the spatial components give the other. Transversality removes the temporal component; vanishing gamma trace excludes the remaining spin-\(1/2\) sector. Four spin-\(3/2\) polarizations remain. When the Dirac equation holds, the condition \(\Gamma^\rho\psi_\rho=0\) already implies \(\partial^\rho\psi_\rho=0\), so the three conditions in (41) are not all independent. They are written together to identify the physical sector explicitly.
Spin 2. Four symmetric Dirac indices correspond to a description using a symmetric tensor \(h_{\lambda\rho}\). The free massive Fierz–Pauli sector [6] is characterized by
\[\tag{42} h_{\lambda\rho}=h_{\rho\lambda},\qquad \left(\Box+\frac{m^2c^2}{\hbar^2}\right)h_{\lambda\rho}=0,\qquad \partial^\lambda h_{\lambda\rho}=0,\qquad \eta^{\lambda\rho}h_{\lambda\rho}=0. \] In the rest frame, a symmetric spatial tensor with six components remains; the tracelessness condition leaves five polarizations. Spin 2 alone does not turn such a free massive field into a complete nonlinear theory of gravity.
These examples thus give 1, 2, 3, 4, and 5 polarizations for massive spins 0, 1/2, 1, 3/2, and 2, respectively. Introducing interactions requires a separate consistency analysis; the free equations do not determine them automatically.
10. The massless limit and the photon
For \(m=0\), there is no rest frame. The proof above based on a transformation to rest therefore does not apply: physical states must be classified by helicity, the angular momentum projection along the momentum direction. For ordinary massless fields of finite helicity, the physical modes also depend on the constraints and gauge structure.
The electromagnetic field illustrates this. In the absence of sources, Maxwell equations written in terms of the potential take the form
\[\tag{43} \partial_\lambda F^{\lambda\rho}=0,\qquad A^\lambda\longmapsto A^\lambda+\partial^\lambda\chi. \] In the Lorenz gauge \(\partial_\lambda A^\lambda=0\), we obtain \(\Box A^\rho=0\). For a plane wave \(A^\lambda=\epsilon^\lambda e^{-ik_\rho x^\rho}\), this implies
\[\tag{44} k_\lambda k^\lambda=0,\qquad k_\lambda\epsilon^\lambda=0,\qquad \epsilon^\lambda\sim\epsilon^\lambda+\zeta k^\lambda. \] The last relation identifies polarizations that differ by a gauge term. For \(k^\lambda=(\kappa,0,0,\kappa)\) with \(\kappa>0\), transversality gives \(\epsilon^0=\epsilon^3\), and a gauge transformation can remove both components. Two transverse directions remain; their circular combinations have helicities \(+1\) and \(-1\).
The Lorenz gauge retains residual freedom satisfying \(\Box\chi=0\); this is the freedom used in (44) to remove the component proportional to \(k^\lambda\). For \(m>0\), the Proca field has no such gauge freedom, and the longitudinal polarization is physical. It cannot simply be declared a gauge mode at an arbitrarily small nonzero mass. The limit of a family of massive solutions and the definition of the physical state space at strictly zero mass require separate consideration.
A massive vector therefore has three polarizations, while the free Maxwell field has two. Simply setting \(m=0\) in the dispersion relation does not explain this change: the newly available gauge freedom must be taken into account, and the physical state space must be selected. Similarly, a massless Dirac spinor retains its spinor transformation law and does not become a photon.
The product formula involving \(J\) remains useful for the corresponding rotation representation. For a photon, however, the three states of massive spin 1 cannot all be treated as physical polarizations. The projection onto physical modes depends on momentum, and rotations also change the momentum direction.
11. Comparison of the results
In the table, N refers to the minimal fully symmetric construction of spin space. The polarization count is for a free massive particle at fixed momentum on one energy branch.
| Spin s | N = 2s | Projections μ | Polarizations | R(2π) | Free massive dynamics |
|---|---|---|---|---|---|
| 0 | 0; a two-factor singlet is another realization | 0 | 1 | +I | Klein–Gordon |
| 1/2 | 1 | ±1/2 | 2 | −I | Dirac |
| 1 | 2 | 1, 0, −1 | 3 | +I | Proca |
| 3/2 | 3 | 3/2, 1/2, −1/2, −3/2 | 4 | −I | Rarita–Schwinger |
| 2 | 4 | 2, 1, 0, −1, −2 | 5 | +I | Fierz–Pauli |
The calculations show that binary selections and exchanges can reproduce representations of any integer or half-integer spin: several factors are combined, the required sector is selected, and its rotation is expressed as a product of elementary \(J\) operators. For free massive fields, this structure is consistent with the Bargmann–Wigner system, whose Dirac operators can likewise be realized through splittings.
The physical selection of a sector remains open. The full tensor product contains several spins, and the algebraic decomposition alone does not explain why a particular one is realized. This selection may be determined by constraints or by the dynamics of the model, but the corresponding law must be formulated separately. Masses, particle statistics, and interactions are not derived here either.
We have obtained a general mathematical method for constructing spin and rotations from splittings. Interpreting these operations as a physical mechanism underlying the internal structure of a particle remains a separate hypothesis of the model.
References
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- H. Rauch et al. Verification of Coherent Spinor Rotation of Fermions. Physics Letters A 54 (1975), 425–427.
- W. Rarita, J. Schwinger. On a Theory of Particles with Half-Integral Spin. Physical Review 60 (1941), 61.
- M. Fierz, W. Pauli. On relativistic wave equations for particles of arbitrary spin in an electromagnetic field. Proceedings of the Royal Society A 173 (1939), 211–232.