Summary: Spatial reflection symmetries force degenerate Dirac polarization multiplets even for unequal scalar confinement axes, and identify when those families also preserve the nonlinear moment source.

Anisotropic scalar confinement can retain polarization degeneracy

The spatial polarization waves use the rest-polarization family of a radial Dirac profile. A stronger statement applies to scalar potentials which are even in each principal spatial coordinate: unequal confinement strengths do not by themselves remove the entire degeneracy.

Work in the stated flat-coordinate spatial model, with a self-adjoint operator and a domain preserved by the reflections below. The bound coordinates are ordinary macroscopic directions of the physical sector. A curved global sector or different boundary conditions require their own symmetry check; a coordinate expression alone does not establish these global symmetries there.

Let

H0=−i∑i=1dαi∂i+βM(x),M(x1,…,−xi,…,xd)=M(x).(1) H_0=-i\sum_{i=1}^d\alpha_i\partial_i+\beta M(x), \qquad M(x_1,\ldots,-x_i,\ldots,x_d)=M(x). \tag{1}

The matrices are Hermitian and satisfy {αi,αj}=2δijI\{\alpha_i,\alpha_j\}=2\delta_{ij}I, {αi,β}=0\{\alpha_i,\beta\}=0, and β2=I\beta^2=I. A scalar quadratic source with a real symmetric coefficient matrix can be written in this form after choosing its principal axes.

Combine each actual coordinate reflection with its Clifford matrix

Define PiΨ(x)=Ψ(Rix)\mathcal P_i\Psi(x)=\Psi(R_ix), where RiR_i reverses the iith coordinate, and put

Ci=αiPi.(2) \boxed{C_i=\alpha_i\mathcal P_i.} \tag{2}

These are unitary Hermitian operators on the spatial field. The coordinate reflections commute with the constant Clifford matrices, so

Ci2=I,CiCj=−CjCi(i≠j).(3) C_i^2=I,\qquad C_iC_j=-C_jC_i\quad(i\ne j). \tag{3}

Conjugating the kinetic operator by CiC_i reverses every term: the coordinate reflection reverses the iith derivative, while αi\alpha_i reverses the other Clifford matrices. The even scalar mass changes sign because αiβαi=−β\alpha_i\beta\alpha_i=-\beta. Hence

CiH0Ci=−H0,[H0,CiCj]=0.(4) \boxed{C_iH_0C_i=-H_0,\qquad [H_0,C_iC_j]=0.} \tag{4}

Every even product preserves the same energy eigenspace. These operators act on actual spatial coordinates and field components; their labels do not describe another physical space.

An identity potential V0(x)IV_0(x)I which is also even in every coordinate commutes with each CiC_i. It therefore preserves the second identity in (4) for H=H0+V0IH=H_0+V_0I, although the first identity then need not hold. The multiplicity argument only needs the even products to commute with HH.

Derive the minimum multiplicity without an assumed particle label

For m=⌊d/2⌋m=\lfloor d/2\rfloor, define commuting Hermitian involutions

Wj=−iC2j−1C2j,j=1,…,m.(5) W_j=-iC_{2j-1}C_{2j},\qquad j=1,\ldots,m. \tag{5}

Each commutes with HH. In a finite isolated energy eigenspace, choose a joint eigenvector with signs wj=±1w_j=\pm1.

For odd d=2m+1d=2m+1, the energy-preserving operator C2jCdC_{2j}C_d flips wjw_j and leaves every other sign unchanged. Its unitarity keeps the result nonzero. Applying these operators produces all 2m2^m sign patterns at the same energy. Different sign patterns are orthogonal.

For even d=2md=2m, use C2jC2mC_{2j}C_{2m} for j<mj<m. It flips wjw_j and wmw_m together, producing 2m−12^{m-1} orthogonal patterns with fixed total sign product. Therefore every such eigenspace contains a multiplet of size at least

Nd=2⌊(d−1)/2⌋.(6) \boxed{N_d=2^{\lfloor(d-1)/2\rfloor}.} \tag{6}

Repeated multiplets can make the degeneracy larger. The minimum sizes at the working depths are:

spatial depth minimum scalar-source multiplet
33 22
55 44
66 44
1010 1616

The proof permits unequal strengths along every axis. It requires the reflection symmetries and operator domain, rather than spherical symmetry. These numbers count degenerate field modes, not additional spatial coordinates or a physical particle census.

For the central five-dimensional family, the balanced total spatial rotation has two neutral members within its four-mode family, as derived in Rotation-neutral Dirac fields. Scalar confinement with the symmetries above cannot create an isolated two-mode energy band by splitting a four-mode multiplet. Occupation may depend on formation and other source terms, which require their own derivation.

Check the nonlinear source before declaring a degenerate solution family

A degeneracy of a frozen operator alone is insufficient for a nonlinear field equation. Different eigenvectors could change the source which was used to construct that operator. For the centered quadratic moment kernel, the relevant source data can be checked explicitly.

Suppose a finite eigenspace consists of exactly one minimal multiplet. For any operator commuting with all the even reflection transformations, its compression onto that multiplet is scalar. This follows directly from the sign proof: commuting with the WjW_j makes it diagonal in the distinct sign states, and commuting with their flips makes all diagonal entries equal.

Every diagonal scalar moment operator βxi2\beta x_i^2 commutes with the even transformations. Its expectation is therefore independent of the chosen normalized polarization in that eigenspace.

For a pure scalar quadratic source

M(x)=m+∑iaixi2,(7) M(x)=m+\sum_i a_ix_i^2, \tag{7}

assume the aia_i are distinct. The full ordinary spatial rotation generator

Jij=−i(xi∂j−xj∂i)−i2αiαj(8) J_{ij}=-i(x_i\partial_j-x_j\partial_i) -\frac i2\alpha_i\alpha_j \tag{8}

obeys

[Jij,H0]=2i(ai−aj)βxixj.(9) \boxed{[J_{ij},H_0] =2i(a_i-a_j)\beta x_ix_j.} \tag{9}

The orbital and spin commutators cancel in the kinetic term. Between any two vectors at the same energy, the left side has zero matrix element. Thus the mixed scalar moment operator has zero compression onto that whole eigenspace.

The parity operator Π=βP1⋯Pd\Pi=\beta\mathcal P_1\cdots\mathcal P_d also commutes with H0H_0 and the even transformations. A minimal multiplet has one parity sign. Its scalar density is even under x↦−xx\mapsto-x, so its scalar first moment vanishes. Provided the total scalar integral is nonzero, its scalar center is the origin.

Consequently every normalized member has the same centered scalar moment matrix. If one solves the nonlinear quadratic moment source equation, all members of that minimal multiplet solve the same source equation. This statement concerns a kernel determined by those moments; another nonlinearity depending on further local density data needs another check.

When some axis coefficients coincide, (9) alone does not determine the corresponding mixed moments. The explicit central and two-block fields in Scalar Dirac modes and Nested Dirac pair separately verify their polarization-independent scalar sources.

Implication for coupling calculations

The scalar model can bind spatial profiles, deform a nested pair and produce polarization-dependent dynamical shape response. Its persistent polarization families explain why those results do not yet select an occupied neutral mode. Increasing axis anisotropy while preserving the same scalar reflection symmetries does not supply that missing energy splitting.

The full shape response can still distinguish prepared polarizations through its angular spectrum and source matrix. The required physical calculation must retain that response and determine which polarization is occupied and which current the incoming field samples. A source symmetry and a transition selection rule are separate parts of that calculation.

Verification

dirac_reflection_symmetry_checks.py checks (4) by differentiating complete spatial fields at coordinate points, including an even nonquadratic scalar potential. An independent Cartesian Hermite regulator with unequal axis coefficients preserves the reflections and reproduces the predicted multiplet sizes at depths three, five, six and ten. These spectra verify multiplicity, not converged physical masses.

The diagnostic checks each sign-flipping operation, the complete orbital-plus-spin commutator, scalar moment compression and parity within one minimal eigenspace. It also verifies that the multiplicity requirement is conditional on the source symmetry. All conditions remain active under Python optimization.