Scalar envelope of a bound Dirac field

Summary: Readjusting the scalar bound profile gives a normalized Dirac envelope potential with a derived density power, exact uniform moving branches and a specified reciprocal fluctuation spectrum.

Derive the envelope interaction from the spatial source

Consider the scalar bound profile and one free normal spatial coordinate in Normal propagation of spatial Dirac modes. The envelope describes amplitudes of the same field on that existing coordinate. It is not another physical space or a mediator particle.

Use a fixed rest polarization and center, and let the real bound radial profile adjust to a positive scalar envelope density $s$. The construction below is a variational reduction to that profile family. Uniform fields give exact solutions of the stated scalar model. Varying envelopes can also drive omitted shape, center and polarization modes; the reduction does not establish their decoupling or the stability of the full field.

Let $\chi(t,y)$ be the two amplitudes of the bound mode and its normal Dirac partner. In their normalized basis,

$$\alpha_{\rm eff}=\sigma_1,\qquad \beta_{\rm eff}=\sigma_3,\qquad s=\chi^\dagger\beta_{\rm eff}\chi>0. \tag{1}$$

The microscopic scalar density is $s\,\phi_s(x)^\dagger\beta\phi_s(x)$, where $\int\phi_s^\dagger\phi_s=1$. These component matrices arise from the full normal partner; they do not replace the bound spatial coordinates.

The massless-background profile fixes the density power

Set the bare scalar coefficient $m=0$ in the bound-profile calculation. For a quadratic source of strength $a$, its positive frequency and unit-profile scalar moment are

$$M(a)=\epsilon_d a^{1/3},\qquad T(a)=\frac{M(a)}{3a}. \tag{2}$$

The envelope density changes the self-consistency equation to

$$a=\frac{g_0sT(a)}d. \tag{3}$$

Solving it gives

$$\boxed{M(s)=M_*s^{1/5},\qquad M_*=\epsilon_d\left(\frac{g_0\epsilon_d}{3d}\right)^{1/5}.} \tag{4}$$

The power follows from the first-order bound kinetic term and the quadratic scalar moment. It is not an independently chosen envelope nonlinearity. The depth enters the profile coefficient $\epsilon_d$ and the common scalar coefficient in (4).

This branch is defined about positive scalar density. Its spatial size scales as $s^{-1/5}$ and its bound frequency as $s^{1/5}$. As $s$ tends to zero the profile spreads and its bound frequency vanishes. A reduction assuming a fixed isolated bound profile cannot be extended to that zero-density limit without checking the full spatial response.

The bound kinetic contribution is $2sM(s)/3$ and the interaction energy is $sM(s)/6$. Thus their complete reduced potential is

$$\boxed{U(s)=\frac56M_*s^{6/5},\qquad U'(s)=M(s).} \tag{5}$$

The derivative includes the response of the spatial profile. Keeping its width fixed while differentiating would give a different interaction.

The normalized variational equation

For the chosen real normalized profile family, the projected connection from varying $s$ vanishes: its diagonal inner product is the derivative of the unit norm and its off-diagonal normal-partner inner product vanishes by angular integration. The restricted first-order action is

$$\boxed{L_{\rm eff} =\int dy\left[ i\chi^\dagger\partial_t\chi -\chi^\dagger(-i\alpha_{\rm eff}\partial_y)\chi -U(\chi^\dagger\beta_{\rm eff}\chi)\right].} \tag{6}$$

It gives

$$i\partial_t\chi =\left[-i\alpha_{\rm eff}\partial_y +\beta_{\rm eff}M(s)\right]\chi. \tag{7}$$

The fixed-center and fixed-polarization restrictions in this action must remain explicit. The full spatial equation can have components outside this profile family when $s$ varies. Eliminating those components requires their actual response and the corresponding dressed vertices; the absence of a projected connection is not a proof that the complement is dynamically inactive.

The omitted term can be calculated explicitly. Let $W_s=(\phi_s,\alpha_y\phi_s)$, with spatial inner products understood. Massless profile scaling gives

$$\phi_s(x)=s^{d/10}\phi_1(s^{1/5}x),\qquad \partial_s\phi_s=\frac1{5s} \left(\frac d2+x\cdot\nabla_x\right)\phi_s. \tag{7a}$$

Angular integration and real radial normalization give $W_s^\dagger W_s=I$ and $W_s^\dagger\partial_sW_s=0$. Nevertheless,

$$(\partial_sW_s)^\dagger(\partial_sW_s)=g_{ss}I,\qquad g_{ss}=\|\partial_s\phi_s\|^2>0. \tag{7b}$$

For a solution of (7), reconstruct $\Psi=W_s\chi$ in the full spatial equation. Since $\alpha_yW_s=W_s\sigma_1$, its residual is

$$\boxed{\mathcal R =(i\partial_t+i\alpha_y\partial_y-H_b)\Psi =i(\partial_sW_s) \big[(\partial_ts)I+(\partial_ys)\sigma_1\big]\chi.} \tag{7c}$$

It obeys

$$W_s^\dagger\mathcal R=0,\qquad \|\mathcal R\|^2 =g_{ss}\left\|\big[(\partial_ts)I +(\partial_ys)\sigma_1\big]\chi\right\|^2. \tag{7d}$$

Thus the envelope equation can hold exactly while a nonzero residual drives the omitted spatial profile. Constant scalar density makes this residual vanish, including the uniform moving solutions below. Small density gradients make the explicit source small, but a controlled elimination still needs the complement's spectrum and response. Its unstable or gapless modes cannot be declared harmless from the vanishing projected connection alone.

Uniform propagation with laboratory density held fixed

For a positive-frequency plane wave with normal momentum $k$, write $n=\chi^\dagger\chi$. Its normalized spinor gives

$$E^2=M(s)^2+k^2,\qquad s=\frac{nM(s)}E. \tag{8}$$

Using (4), these relations are equivalent to

$$\boxed{M^4E=M_*^5n,\qquad E(E^2-k^2)^2=M_*^5n,\qquad E>|k|.} \tag{9}$$

There is one positive solution: the left side increases from zero for $E>|k|$. At fixed laboratory density, changing momentum changes the scalar density and therefore the bound profile. This is consistent with the normal boost calculation, where holding the proper scalar density fixed instead gives a constant $M$ and a laboratory density $n=sE/M$.

The total energy density is

$$\boxed{\mathcal H(n,k)=nE-\frac{nM^2}{6E}.} \tag{10}$$

The subtracted term is the interaction counted twice in density times the nonlinear phase frequency. Independent variation gives

$$\frac{\partial\mathcal H}{\partial n}=E,\qquad \frac{\partial\mathcal H}{\partial k}=\frac{nk}{E}=j_y. \tag{11}$$

Thus the action fixes the frequency, current and scalar source together. For a nonlinear plane family, the derivative of its phase frequency at fixed density is not a substitute for the energy derivative in (11).

The normal stress has a consistent Lorentz transformation

At fixed proper scalar density $s$, define the rest energy density and pressure of this uniform reduced field by

$$\varepsilon_0=U(s)=\frac56sM, \qquad P=sU'(s)-U(s)=\frac16sM. \tag{12}$$

For $\gamma=E/M$ the transformed energy is

$$\gamma^2(\varepsilon_0+P)-P =nE-\frac{nM^2}{6E}, \tag{13}$$

exactly (10). The field has a pressure as well as an energy density. It is not the kinematics of one noninteracting point particle with a fixed energy per unit laboratory length. These are properties of the specified normal envelope model, with no electric charge assigned.

A complete fluctuation check within the envelope model

At rest take $\chi_0=\sqrt n\,(1,0)^T e^{-iMt}$, with $M=M_*n^{1/5}$. The scalar source derivative is $a=nM'(n)=M/5$. For a normal Fourier wavenumber $q$, the ordinary and conjugate fluctuation matrix is

$$\mathcal B(q)= \begin{pmatrix} a&q&a&0\\ q&-2M&0&0\\ -a&0&-a&q\\ 0&0&q&2M \end{pmatrix}. \tag{14}$$

The conjugate row contains the opposite Fourier momentum before complex conjugation; its displayed kinetic signs retain that operation. Direct evaluation of the characteristic polynomial gives

$$\boxed{\omega_\pm^2(q) =q^2+2M^2\pm2M\sqrt{M^2+\frac45q^2}.} \tag{15}$$

Both values are nonnegative. Their product is $q^4+(4/5)M^2q^2$ and their sum is positive. The low branch satisfies $\omega_-^2=q^2/5+O(q^4/M^2)$, agreeing with $dP/d\varepsilon_0=1/5$ from (12).

This verifies normal fluctuations of the declared reduced action. It does not establish stability of the bound field: the $m=0$ radial branches at several working depths have the separate growing modes already documented in the bound-profile note. Those degrees of freedom, together with changing centers and polarizations, remain part of a full coupling calculation.

Reciprocal scalar envelopes readjusts two nested spatial profiles together and derives their full scalar coupling matrix from one common interaction coefficient.

Spatial Dirac polarization waves restores the other rest polarizations. Their linear normal waves have a quadratic gapless branch, and certain finite polarized superpositions preserve the scalar source exactly while their bound spatial current changes.

Verification

normal_dirac_transport_checks.py solves (9) at several densities and momenta, independently reconstructs the adjusted bound profiles and checks their scalar source equations. Finite differences of the full energy verify both derivatives in (11). The normal Lorentz stress is checked against that same energy, and the complete ordinary/conjugate matrix spectrum is compared with (15). Independent finite differences of the full spatial field verify (7c)–(7d) for complex envelope amplitudes, including the nonzero residual norm and its zero projection onto the retained amplitudes. All conditions remain active under Python optimization.

Related

• Normal propagation of spatial Dirac modes — provides the full spatial boost and the density convention needed here.
• Self-consistent scalar Dirac moment modes — derives the profile scaling and its full radial stability limits.
• Dressed spatial source vertices — supplies the reconstruction required if omitted modes are eliminated.