In Models of Spontaneous Wave-Function Collapse, Why Only Fermions Collapse, Not Bosons?

1. Introduction

A recurring conceptual obstacle in quantum gravity is the role of time: standard quantum theory (and, in particular, quantum field theory) is formulated with respect to an external classical time parameter, whereas general relativity treats spacetime geometry as dynamical. Generalized trace dynamics (GTD) [1] addresses this tension by building on Adler’s trace dynamics (TD) [2], a deterministic matrix-valued Lagrangian/Hamiltonian dynamics in which the action is the trace of a polynomial in noncommuting matrix degrees of freedom (bosonic even-grade matrices and fermionic odd-grade matrices). A key structural feature of TD is its global unitary invariance, which gives rise to a novel conserved quantity—the Adler–Millard charge—with dimensions of action,

$$\tilde{C}=\sum _{r\in B}[q_r,p_r]-\sum _{r\in F}\{q_r,p_r\},$$

(1)

absent in ordinary classical dynamics. When the statistical mechanics of TD is developed, equipartition of $\tilde{C}$ at equilibrium yields the canonical (anti)commutation relations and an emergent unitary quantum (field) dynamics for the canonical averages. Conversely, when the anti-self-adjoint part of the fundamental trace Hamiltonian is not negligible, fluctuations about equilibrium can drive effective nonunitary nonlinear stochastic dynamics and spontaneous localization [3].

GTD extends this pre-quantum framework toward quantum gravity by promoting gravitational degrees of freedom to matrices and by replacing classical spacetime labels by a noncommutative “pre-spacetime” (quaternionic/octonionic, in the models of interest), thereby aiming for a formulation of quantum dynamics without reference to an external classical time (evolution may be parametrized by Connes time $\tau$) [1]. The fundamental entities are atoms of spacetime-matter (STM atoms): matrix degrees of freedom $q_i=q_{Bi}+q_{Fi}$ whose bosonic components are interpreted as “atoms of spacetime,” while the full STM atom represents a fermion together with the bosonic fields it sources (for example, an electron together with its electromagnetic, weak, and gravitational fields). Entanglement among many STM atoms, followed by spontaneous localization, is conjectured to yield classical spacetime geometry and classical macroscopic bodies [4]. In this setting, objective collapse is expected precisely in regimes where the fundamental trace Hamiltonian develops a significant anti-self-adjoint component. This motivates the specific structural question addressed below: whether the STM-atom trace Lagrangian underlying GTD generates an intrinsic anti-self-adjoint contribution only in the fermionic sector—thereby explaining why collapse models effectively act on fermions and not on bosonic degrees of freedom.

Spontaneous collapse models (e.g. CSL-type modifications of quantum dynamics) [5,6,7,8,9] are commonly set up so that the collapse-inducing nonunitarity couples to matter degrees of freedom, while bosonic field degrees do not constitute independent collapse channels. From the viewpoint of generalized trace dynamics (GTD), a natural structural origin of collapse is the appearance of an anti-self-adjoint (ASA) component of the trace Hamiltonian: when ASA effects are significant, coarse-graining can yield effective nonunitary stochastic dynamics and spontaneous localization (as in Adler-type trace dynamics arguments [2]).

A central open issue for GTD phenomenology is then:

Does the fundamental GTD STM-atom Lagrangian generate ASA contributions only in the fermionic sector, thereby explaining why only fermions collapse?

In this note we answer this question in the affirmative, for the STM-atom action written in terms of two inequivalent matrix degrees of freedom $Q_1$ and $Q_2$. We work directly at the level of the trace Lagrangian and its Legendre transform, using trace derivatives in the sense of Adler, and we keep variations with respect to bosonic and fermionic variables strictly separated.

2. STM-Atom Action and Unified Variables

2.1. Action

We begin with the unified STM-action [1]

$${\displaystyle \frac{S}{\hslash }}={\displaystyle \frac{1}{2}}\int {\displaystyle \frac{d\tau }{{\tau }_{\mathrm{Pl}}}}\mathrm{Tr}\left[\lambda {\dot{Q}}_1^{†}{\dot{Q}}_2\right],\lambda :={\displaystyle \frac{L_{\mathrm{Pl}}^2}{L^2}}.$$

(2)

Here $\tau$ is Connes time and † denotes the GTD adjoint on matrices with Grassmann entries.

The unified velocities are defined (with bosonic/fermionic splitting) by

$$\begin{array}{cccc}\hfill {\dot{Q}}_B& ={\displaystyle \frac{1}{L}}\left(i\alpha q_B+L{\dot{q}}_B\right),\hfill & \hfill {\dot{Q}}_F& ={\displaystyle \frac{1}{L}}\left(i\alpha q_F+L{\dot{q}}_F\right),\hfill \end{array}$$

(3)

$$\begin{array}{cccc}\hfill {\dot{Q}}_1^{†}& ={\dot{Q}}_B^{†}+\lambda {\beta }_1{\dot{Q}}_F^{†},\hfill & \hfill {\dot{Q}}_2& ={\dot{Q}}_B+\lambda {\beta }_2{\dot{Q}}_F,\hfill \end{array}$$

(4)

where ${\beta }_1,{\beta }_2$ are odd-grade Grassmann elements inserted so that the trace Lagrangian is bosonic. The dynamics requires ${\beta }_1\ne {\beta }_2$[10]. Also, $\alpha$ is a real number which stands for the Yang-Mills coupling constant.

2.2. Grading, Adjoint, and Graded Cyclicity

We write $\epsilon \left(X\right)\in \{0,1\}$ for the Grassmann parity: $\epsilon \left(X\right)=0$ for bosonic (even-grade) matrices and $\epsilon \left(X\right)=1$ for fermionic (odd-grade) matrices. We assume

$$\epsilon \left({\dot{Q}}_B\right)=0,\epsilon \left({\dot{Q}}_F\right)=1,\epsilon \left({\beta }_a\right)=1(a=1,2).$$

(5)

Thus ${\beta }_a{\dot{Q}}_F$ and ${\beta }_a{\dot{Q}}_F^{†}$ are even, so ${\dot{Q}}_1^{†}$ and ${\dot{Q}}_2$ are bosonic as required by the form of the trace action.

For matrices with Grassmann entries one uses the standard involution

$${\left(AB\right)}^{†}=B^{†}{A}^{†},\mathrm{Tr}{\left(A\right)}^{†}=\mathrm{Tr}\left(A^{†}\right),$$

(6)

together with graded cyclicity of the trace for homogeneous factors:

$$\mathrm{Tr}\left(AB\right)={(-1)}^{\epsilon \left(A\right)\epsilon \left(B\right)}\mathrm{Tr}\left(BA\right).$$

(7)

(Equivalently, a cyclic shift of a homogeneous factor X through a product Y yields $\mathrm{Tr}\left(YX\right)={(-1)}^{\epsilon \left(X\right)\epsilon \left(Y\right)}\mathrm{Tr}\left(XY\right)$.)

3. Trace-Derivative Canonical Momenta and Hamiltonian

3.1. Trace Lagrangian

From (2) the trace Lagrangian density (in $\tau$) is

$$\mathcal{L}={\displaystyle \frac{\hslash }{2{\tau }_{\mathrm{Pl}}}}\mathrm{Tr}\left[\lambda {\dot{Q}}_1^{†}{\dot{Q}}_2\right].$$

(8)

Crucially, $\mathcal{L}$ depends on the bosonic velocities ${\dot{Q}}_B,{\dot{Q}}_B^{†}$ and fermionic velocities ${\dot{Q}}_F,{\dot{Q}}_F^{†}$ only through ().

3.2. Trace Derivatives (Bosons and Fermions Varied Separately)

Following Adler, the trace derivative is defined by placing the variation to the far right:

$$\delta \mathcal{L}=\mathrm{Tr}\left({\displaystyle \frac{\delta \mathcal{L}}{\delta O}}\delta O\right),$$

(9)

where for fermionic O the graded cyclicity (7) is used to move $\delta O$ to the right, generating sign factors only when $\delta O$ passes odd-grade factors. In the present computation we do not vary with respect to mixed variables; we only vary with respect to ${\dot{Q}}_B,{\dot{Q}}_B^{†}$ (bosonic) and ${\dot{Q}}_F,{\dot{Q}}_F^{†}$ (fermionic).

3.3. Canonical Momenta

Define the overall prefactor

$$c:={\displaystyle \frac{\hslash }{2{\tau }_{\mathrm{Pl}}}}.$$

(10)

3.3.0.1. Bosonic momenta.

Varying (8) with respect to ${\dot{Q}}_B$ and ${\dot{Q}}_B^{†}$ gives

$$\begin{array}{cc}\hfill {\Pi }_B& :={\displaystyle \frac{\delta \mathcal{L}}{\delta {\dot{Q}}_B}}=c\lambda {\dot{Q}}_1^{†},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\Pi }_B^{†}& :={\displaystyle \frac{\delta \mathcal{L}}{\delta {\dot{Q}}_B^{†}}}=c\lambda {\dot{Q}}_2.\hfill \end{array}$$

(12)

3.3.0.2. Fermionic momenta.

Using ${\dot{Q}}_2={\dot{Q}}_B+\lambda {\beta }_2{\dot{Q}}_F$,

$${\Pi }_F:={\displaystyle \frac{\delta \mathcal{L}}{\delta {\dot{Q}}_F}}=c\lambda {\dot{Q}}_1^{†}·\left(\lambda {\beta }_2\right)=c{\lambda }^2{\dot{Q}}_1^{†}{\beta }_2.$$

(13)

Similarly, using ${\dot{Q}}_1^{†}={\dot{Q}}_B^{†}+\lambda {\beta }_1{\dot{Q}}_F^{†}$ and bringing $\delta {\dot{Q}}_F^{†}$ to the right inside the trace using (7),

$${\Pi }_F^{†}:={\displaystyle \frac{\delta \mathcal{L}}{\delta {\dot{Q}}_F^{†}}}=c{\lambda }^2{\dot{Q}}_2{\beta }_1.$$

(14)

Equations (11)–(14) are the separated bosonic/fermionic canonical momenta required for a trace-dynamics Legendre transform.

3.4. Legendre Transform and Explicit Hamiltonian

We define the trace Hamiltonian by summing over all independent velocities:

$$\mathcal{H}=\mathrm{Tr}\left({\Pi }_B{\dot{Q}}_B+{\Pi }_F{\dot{Q}}_F+{\Pi }_B^{†}{\dot{Q}}_B^{†}+{\Pi }_F^{†}{\dot{Q}}_F^{†}\right)-\mathcal{L}.$$

(15)

Substituting (11)–(14) and using (), one finds

$$\begin{array}{cc}\hfill \mathrm{Tr}({\Pi }_B{\dot{Q}}_B+{\Pi }_F{\dot{Q}}_F)& =c\mathrm{Tr}\left(\lambda {\dot{Q}}_1^{†}({\dot{Q}}_B+\lambda {\beta }_2{\dot{Q}}_F)\right)=c\mathrm{Tr}\left(\lambda {\dot{Q}}_1^{†}{\dot{Q}}_2\right)=\mathcal{L},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \mathrm{Tr}({\Pi }_B^{†}{\dot{Q}}_B^{†}+{\Pi }_F^{†}{\dot{Q}}_F^{†})& =c\mathrm{Tr}\left(\lambda ({\dot{Q}}_B^{†}+\lambda {\beta }_1{\dot{Q}}_F^{†}){\dot{Q}}_2\right)=c\mathrm{Tr}\left(\lambda {\dot{Q}}_1^{†}{\dot{Q}}_2\right)=\mathcal{L},\hfill \end{array}$$

(17)

where in the second line we used that ${\dot{Q}}_2$ is bosonic and applied graded cyclicity. Therefore

$$\begin{array}{|c|}\hline \mathcal{H}=\mathcal{L}={\displaystyle \frac{\hslash }{2{\tau }_{\mathrm{Pl}}}}\mathrm{Tr}\left[\lambda {\dot{Q}}_1^{†}{\dot{Q}}_2\right].\\ \hline\end{array}$$

(18)

This is the trace-dynamics analog of the Bateman-type cross-kinetic structure: the Hamiltonian equals the Lagrangian, but need not be self-adjoint because ${\dot{Q}}_1$ and ${\dot{Q}}_2$ are inequivalent.

4. Bosonic vs Fermionic Contributions and the Anti-Self-Adjoint Part

4.1. Decomposition of the Hamiltonian

Expanding (18) using () gives

$$\mathcal{H}={\mathcal{H}}_{BB}+{\mathcal{H}}_{BF}+{\mathcal{H}}_{FF},$$

(19)

where

$$\begin{array}{cc}\hfill {\mathcal{H}}_{BB}& =c\mathrm{Tr}\left(\lambda {\dot{Q}}_B^{†}{\dot{Q}}_B\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\mathcal{H}}_{BF}& =c\mathrm{Tr}\left({\lambda }^2{\dot{Q}}_B^{†}{\beta }_2{\dot{Q}}_F+{\lambda }^2{\beta }_1{\dot{Q}}_F^{†}{\dot{Q}}_B\right),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\mathcal{H}}_{FF}& =c\mathrm{Tr}\left({\lambda }^3{\beta }_1{\dot{Q}}_F^{†}{\beta }_2{\dot{Q}}_F\right).\hfill \end{array}$$

(22)

4.2. Adjoint Assumptions for ${\beta }_1,{\beta }_2$

We now impose the following adjoint properties:

$$\begin{array}{|c|}\hline {\beta }_1^{†}=-{\beta }_1,{\beta }_2^{†}=-{\beta }_2,{\beta }_1\ne {\beta }_2,\epsilon \left({\beta }_1\right)=\epsilon \left({\beta }_2\right)=1.\\ \hline\end{array}$$

(23)

We also assume the natural graded (anti)commutation with dynamical variables: ${\beta }_a$ commutes with bosonic variables and anticommutes with fermionic variables.

4.3. Self-Adjointness of the Purely Bosonic Sector

From (20) and (6),

$${\mathcal{H}}_{BB}^{†}=c\mathrm{Tr}\left(\lambda {\left({\dot{Q}}_B^{†}{\dot{Q}}_B\right)}^{†}\right)=c\mathrm{Tr}\left(\lambda {\dot{Q}}_B^{†}{\dot{Q}}_B\right)={\mathcal{H}}_{BB}.$$

(24)

Hence the bosonic subsector has a self-adjoint Hamiltonian and does not by itself supply an ASA contribution.

4.4. An Intrinsic ASA Fermionic Contribution

Consider the fermionic term (). Using (6) and (23),

$$\begin{array}{cc}\hfill {\mathcal{H}}_{FF}^{†}& =c\mathrm{Tr}\left({\left({\lambda }^3{\beta }_1{\dot{Q}}_F^{†}{\beta }_2{\dot{Q}}_F\right)}^{†}\right)=c\mathrm{Tr}\left({\lambda }^3{\dot{Q}}_F^{†}{\beta }_2^{†}{\dot{Q}}_F{\beta }_1^{†}\right)\hfill \\ \hfill & =c\mathrm{Tr}\left({\lambda }^3{\dot{Q}}_F^{†}(-{\beta }_2){\dot{Q}}_F(-{\beta }_1)\right)=c\mathrm{Tr}\left({\lambda }^3{\dot{Q}}_F^{†}{\beta }_2{\dot{Q}}_F{\beta }_1\right).\hfill \end{array}$$

(25)

Now apply graded cyclicity (7) to cyclically shift the final odd factor ${\beta }_1$ to the front. Since ${\beta }_1$ is odd and the product ${\dot{Q}}_F^{†}{\beta }_2{\dot{Q}}_F$ is also odd (three odd factors), we obtain a minus sign:

$$\mathrm{Tr}\left({\dot{Q}}_F^{†}{\beta }_2{\dot{Q}}_F{\beta }_1\right)=-\mathrm{Tr}\left({\beta }_1{\dot{Q}}_F^{†}{\beta }_2{\dot{Q}}_F\right).$$

(26)

Combining (25) and (26) yields

$$\begin{array}{|c|}\hline {\mathcal{H}}_{FF}^{†}=-{\mathcal{H}}_{FF}.\\ \hline\end{array}$$

(27)

Thus ${\mathcal{H}}_{FF}$ is purely anti-self-adjoint. In particular, it provides an explicit ASA contribution to the Hamiltonian, and this contribution vanishes identically when the fermionic sector is absent.

4.4.0.3. Role of ${\beta }_1\ne {\beta }_2$.

If one attempted to set ${\beta }_1={\beta }_2=\beta$, the fermionic contribution collapses: because $\beta$ is odd, $\beta {\dot{Q}}_F^{†}\beta =-\beta \beta {\dot{Q}}_F^{†}=0$, so ${\mathcal{H}}_{FF}$ would vanish. Hence the requirement ${\beta }_1\ne {\beta }_2$ is not only dynamically necessary [10]; it is also structurally responsible for a nontrivial ASA fermionic term.

4.5. ASA Part of the Full Hamiltonian

Define the self-adjoint and anti-self-adjoint parts of the trace Hamiltonian by

$${\mathcal{H}}_{\mathrm{sa}}:={\displaystyle \frac{1}{2}}(\mathcal{H}+{\mathcal{H}}^{†}),{\mathcal{H}}_{\mathrm{asa}}:={\displaystyle \frac{1}{2}}(\mathcal{H}-{\mathcal{H}}^{†}).$$

(28)

From (19), (24), and (27),

$$\begin{array}{|c|}\hline {\mathcal{H}}_{\mathrm{asa}}={\displaystyle \frac{1}{2}}({\mathcal{H}}_{BF}-{\mathcal{H}}_{BF}^{†})+{\mathcal{H}}_{FF}.\\ \hline\end{array}$$

(29)

Independently of the detailed adjoint properties of the mixed term ${\mathcal{H}}_{BF}$, the key point is that every contribution to ${\mathcal{H}}_{\mathrm{asa}}$ contains fermionic variables: if ${\dot{Q}}_F={\dot{Q}}_F^{†}=0$, then ${\mathcal{H}}_{\mathrm{asa}}=0$ identically.

4.5.0.4. Remark (adjoint convention for ${\beta }_{1,2}$).

The anti-self-adjoint character of the purely fermionic contribution ${\mathcal{H}}_{FF}=c\mathrm{Tr}\left({\lambda }^3{\beta }_1{\dot{Q}}_F^{†}{\beta }_2{\dot{Q}}_F\right)$ does not rely on taking ${\beta }_a^{†}=-{\beta }_a$ individually. More generally, assume

$${\beta }_a^{†}={\eta }_a{\beta }_a,{\eta }_a\in \{+1,-1\},a=1,2,$$

with $\epsilon \left({\beta }_a\right)=1$ and the standard GTD involution ${\left(AB\right)}^{†}=B^{†}{A}^{†}$. Then

$$\begin{array}{cc}\hfill {\mathcal{H}}_{FF}^{†}& =c{\lambda }^3\mathrm{Tr}\left({\left({\beta }_1{\dot{Q}}_F^{†}{\beta }_2{\dot{Q}}_F\right)}^{†}\right)=c{\lambda }^3{\eta }_1{\eta }_2\mathrm{Tr}\left({\dot{Q}}_F^{†}{\beta }_2{\dot{Q}}_F{\beta }_1\right)\hfill \\ \hfill & =-c{\lambda }^3{\eta }_1{\eta }_2\mathrm{Tr}\left({\beta }_1{\dot{Q}}_F^{†}{\beta }_2{\dot{Q}}_F\right)=-\left({\eta }_1{\eta }_2\right){\mathcal{H}}_{FF},\hfill \end{array}$$

where the minus sign arises from graded cyclicity when moving the odd factor ${\beta }_1$ past the odd product ${\dot{Q}}_F^{†}{\beta }_2{\dot{Q}}_F$. Hence ${\mathcal{H}}_{FF}$ is purely anti-self-adjoint whenever ${\eta }_1{\eta }_2=+1$ (i.e. both ${\beta }_1,{\beta }_2$ are self-adjoint, or both are anti-self-adjoint). In particular, the choice ${\beta }_1^{†}={\beta }_1$ and ${\beta }_2^{†}={\beta }_2$ yields the same ASA fermionic term (and thus the same collapse channel) as the choice ${\beta }_a^{†}=-{\beta }_a$.

5. Interpretation for Spontaneous Localization

In trace dynamics, coarse-graining around statistical equilibrium yields two distinct regimes: (i) if the Hamiltonian is effectively self-adjoint, one recovers emergent unitary quantum dynamics; (ii) if the anti-self-adjoint component is significant, coarse-graining yields effective nonunitary (stochastic) dynamics with spontaneous localization [3].

The computation above provides a structural mechanism for “fermion-only collapse” in GTD:

• The purely bosonic Hamiltonian ${\mathcal{H}}_{BB}$ is self-adjoint and thus does not generate ASA-driven collapse by itself.
• The fermionic sector carries an intrinsic anti-self-adjoint term ${\mathcal{H}}_{FF}$, present precisely because ${\beta }_1$ and ${\beta }_2$ are unequal odd Grassmann elements inserted to make the unified trace Lagrangian bosonic.

Therefore, in regimes where ASA effects drive localization, the fundamental collapse channel is naturally associated with fermionic degrees of freedom. Bosonic fields can still become classical indirectly, because in the STM-atom picture an STM atom consists of a fermion together with its associated bosonic fields, so localization of the fermionic degrees forces classicality of the associated bosonic sector through correlations/entanglement, without requiring independent bosonic collapse.

6. Conclusions

Starting from the unified STM-atom GTD trace Lagrangian, we computed the trace Hamiltonian via trace-derivative canonical momenta, with bosonic and fermionic variations treated separately. We found:

1.

The trace Hamiltonian equals the trace Lagrangian (Bateman-type cross-kinetic structure), but it need not be self-adjoint because $Q_1$ and $Q_2$ are inequivalent.

2.

The purely bosonic contribution ${\mathcal{H}}_{BB}$ is self-adjoint.

3.

Assuming natural adjoint properties for the odd Grassmann elements ${\beta }_1,{\beta }_2$, the fermionic contribution ${\mathcal{H}}_{FF}$ is purely anti-self-adjoint, providing an explicit ASA component of the Hamiltonian that vanishes in the bosonic subsector.

This provides a first-principles explanation, within GTD, for why spontaneous localization acts fundamentally on fermionic degrees of freedom and not on bosonic degrees of freedom.

6.0.0.5. Outlook.

The next steps are to connect the magnitude of ${\mathcal{H}}_{FF}$ (and fluctuations about equilibrium) to collapse phenomenology (rates, noise kernels, and effective CSL parameters), and to embed the analysis into the multi-STM-atom setting where entanglement and interactions become operative.

Declaration of Use of AI Tools in this Work: During the preparation of this work the author used Open AI’s ChatGPT 5.2 Pro in order to assist in carrying out calculations and assist in manuscript writing. After using this tool/service, the author reviewed and edited the content as needed and takes full responsibility for the content of the published article.