# "Internal times" and how to second-quantize fields by means of periodic boundary conditions

Donatello Dolce

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Donatello Dolce. "Internal times" and how to second-quantize fields by means of periodic boundary conditions. Annals of Physics, 2023, 457, pp.169398. 10.1016/j.aop.2023.169398. hal-05402192

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# “Internal Times” and How to Second-Quantize Fields by means of Periodic Boundary Conditions

Donatello Dolce

*University of Camerino, Piazza Cavour 19F, 62032 Camerino, Italy.*

## Abstract

With simple yet rigorous arguments, we show that the ordinary second quantization of bosonic and fermionic fields is formally equivalent to constraining the classical field modes with periodic boundary conditions in time. This result confirms and extends earlier findings demonstrating that intrinsically periodic classical mechanics, which can be associated with any Hamiltonian system, is formally equivalent to the canonical quantum mechanics of the system itself. Within this framework, foundational structures such as the Hilbert space, the Schrödinger equation, canonical (anti-)commutation relations, and even the Feynman path integral emerge naturally from intrinsically cyclic dynamics, together with other remarkable correspondences of both phenomenological and theoretical interest. For both scalar and spinor fields, the condition of intrinsic time periodicity directly yields the canonical algebra of ladder operators and field commutators, without postulating them as axioms. These results suggest that quantization may be rooted in the non-trivial topology of time, and in concepts akin to de Broglie’s “internal clocks” and Rovelli’s “internal times,” offering new insights into the longstanding problem of quantizing gravity.
# I. INTRODUCTION

Wave-particle duality in Quantum Mechanics (QM) implies that every elementary particle (or, more generally, any elementary physical system) is described by a wave function $\phi(\mathbf{x}, t)$. That is a “periodic phenomenon”, as originally named by de Broglie [1], whose natural temporal recurrence is determined by the energy via the Planck constant (we adopt natural units $c = \hbar = 1$). If the system at a given time $t_x$ has a locally varying energy $\omega(\mathbf{x})$ due to interactions, the wave function acquires a corresponding local frequency $f(\mathbf{x})$, and therefore its time recurrence $T(\mathbf{x})$ is locally modulated:

$$ \phi(\mathbf{x}, t_x) = \phi(\mathbf{x}, t_x + T(\mathbf{x})). \qquad (1) $$

In a perturbative description of interactions, $T(\mathbf{x})$ changes point by point, following the local change of energy in a causal manner, see also undulatory mechanics and Hamilton’s opto-mechanical analogy [2].

In previous works [3–12] we investigated the following natural question: What happens if this local time recurrence $T(\mathbf{x})$ of the wave function $\phi(\mathbf{x}, t)$, associated with any physical system as in eq.(1), is imposed as an exact constraint on the wave function via Periodic Boundary Conditions (PBCs)?

A general answer was obtained in [3] in the form of a mathematical theorem, using results from Geometric Quantization [14–17], without interpretational ambiguities or fine-tuning of parameters. Under the very general assumptions of Hamiltonian systems, i.e. generic symplectic manifolds of even dimension equipped with a non-degenerate closed 2-form and a global $U(1)$ symmetry, imposing such PBCs to the classical dynamics yields a quantization exactly equivalent to the canonical quantization of the system itself. The resulting cyclic dynamics is naturally described in a Hilbert space, with time evolution governed by the Schrödinger equation; in this framework, PBCs (up to a global twist factor) play a role formally equivalent to canonical commutation relations and the other Dirac rules of QM:

$$ \phi(\mathbf{x}, t_x)^{\text{PBCs}} \equiv \phi(\mathbf{x}, t_x + T(\mathbf{x})) \iff [x^i, \hat{p}_j] = i\delta_{ij}, $$

resulting that the momentum operator is $\hat{p}_i = -i \frac{\partial}{\partial x^i}$, with $i, j = 1, 2, 3$.

Stokes’ lemma guarantees that, along the integral curves of the system, the local temporal recurrence $T(\mathbf{x})$ forms vortex tubes in the extended phase space whose circulation is invariance equal to the Planck constant $h$ (Weil integrability condition) [2], in agreement with
the de Broglie-Planck hypothesis and therefore respecting causality. Under the constraint of intrinsic periodicity, a wave function with definite local energy $\omega(\mathbf{x})$ behaves like a vibrating string, supporting an entire (generally anharmonic in the case of interacting systems) quantized spectrum of modes allowed by the constraint — analogous to the quantization of the mass spectrum in Kaluza-Klein theory from intrinsic periodicity in a compact extra dimension.

The theorem proven in [3] thus confirms and generalizes to every possible classical system earlier results obtained independently for elementary particles, both in the Feynman path-integral and canonical formulations [4–12]. The same formalism has been applied to reinterpret quantum phenomena such as superconductivity and graphene physics [11, 12], to reveal a geometrodynamical origin of gauge interactions analogous to gravitation in General Relativity (GR) [5, 7, 37], see par.(IV) for a short review, and to provide a formal derivation of the AdS/CFT correspondence [6].

In the present paper we demonstrate that this exact formal equivalence between intrinsically periodic dynamics and quantum theory applies naturally to Quantum Field Theory (QFT) in terms of second quantization, for both bosonic and fermionic fields [18]. From the perspective of the general correspondence proven in [3], this result is expected since fields are themselves Hamiltonian (symplectic) systems, albeit with infinitely many degrees of freedom (d.o.f.). We establish a direct correspondence between temporal PBCs and the field commutators of standard second quantization, through arguments entirely independent from those of earlier proofs.

Intuitively, the idea can be regarded as a sort of relativistic generalization, along the time dimension, of the semi-classical “particle in a Box” quantization, where (P)BCs are imposed in time rather than in space. Notice that, if imposing PBCs in space raises no conceptual difficulty, the same should hold for PBCs in time. Indeed, Lorentz boosts “mix” space and time coordinates, so that spatial PBCs in one inertial frame generally appear as a combination of spatial and temporal PBCs in another frame. From a relativistic standpoint, therefore, PBCs in time have the same legitimacy as PBCs in space.<fnref n="1" /> The quantization obtained here, however, is proven to be fully equivalent to canonical quantization for any Hamiltonian system, not just a semi-classical approximation. Up to a Wick rotation, it can

<footnote n="1">¹ The idea of applying PBCs along the time dimension has also been explored in the context of time crystals [13].</footnote>
also be interpreted analogously to the dimensional reduction of Euclidean time in finite-
temperature QFT, [11, 12, 19, 20]. The correspondence between classical and quantum
dynamics emerging from intrinsic Euclidean-time periodicity in thermal field theory closely
parallels that obtained from intrinsic Minkowskian-time periodicity in ordinary QFT, as
studied here. A similar dimensional-reduction idea was used in [6] to prove the classical-
quantum correspondence at the base of the AdS/CFT correspondence [21].

This analogy also provides a further physical motivation for our ansatz, particularly in
relation to the problem of time in Quantum Gravity (QG). As Rovelli argued in [22], sta-
tistical mechanics and equilibrium thermodynamics suggest the necessity of “internal times”
when applied to GR. Here, Rovelli’s proposal finds justification directly within QFT and
QM [3]. Elementary Cycles Theory (ECT) [3–12], which underlies the present work, is ex-
plicitly based on the postulate that every elementary particle behaves as an *elementary clock*
of nature, with a rest period $T_C$ determined by its mass $m$, that is to say $T_C = 1/m$, in
close analogy with de Broglie’s “internal clocks”. Being a clock, each particle can intrinsi-
cally defines its own “internal time coordinate”, characterized by Minkowskian periodicity as
the possible origin of quantum behavior. Since these periodicities are locally and causally
modulated by energy exchanges and interactions, and ensembles of modulated clocks obey
statistical laws, this framework naturally points to the emergence of a thermodynamical
time and an entropic arrow of time, as discussed in our previous works.

Since we are interested to second quantization, the discussion doesn't concern about in-
teractions. The proofs presented involves the study of the Harmonic Oscillator (HO), avoid-
ing the complication of describing local modulations of periodicity. In par.(IV) we however
discuss how to extend our results to interacting fields. Among the various approaches, we
briefly review the formalism of locally modulated periodicities fully developed in [5, 37],
where gauge interactions emerge directly from space-time geometrodynamics (*Principle of
General Invariance*), owing to the nontrivial topology of space-time in ECT — the exact na-
ture of the relativistic geometrodynamics originating standard electromagnetism has been
obtained in [37] after the publication of this paper, in terms of Fermi-walker transport.
These results may contribute to a deeper understanding of quantization and to addressing
conceptual issues related to time in the quantization of GR.
## II. SECOND QUANTIZATION OF BOSONS

The quantization of the HO is at the base of the second quantization of fields and it is particularly simple to interpret in terms of intrinsic periodicity. Being an isochronous system, the HO has global periodicity $T$, i.e. the period does not vary along the integral curves.

### A. Harmonic Oscillator

Consider a HO of characteristic fundamental period $T = 2\pi/\omega$ in one spatial dimension. This must be regarded a single d.o.f. of a Klein-Gordon (KG) field. The Hamiltonian is

$$H_{HO} = \frac{1}{2}(p^2 + \omega^2 q^2) = \frac{1}{2}\omega(a^*a + aa^\dagger),$$

where $a$ and $a^*$ parametrize the phase-space variables as usual: $q = \sqrt{\frac{1}{2\omega}}(a + a^*)$ and $p = i\sqrt{\frac{\omega}{2}}(a^* - a)$.

Before starting our proof we note that, for any arbitrary time $t$, the constraint of global intrinsic periodicity $q(t+T) \stackrel{\text{PBCs}}{=} q(t)$ implies that each solution $q(t)$ is a generic superposition of harmonic eigenmodes $q_n(t) = A_n \exp(-i\omega_n t)$ for some normalization constants $A_n$ (Fourier coefficients), such that $i\partial_t q_n(t) = \omega_n q_n(t)$, with harmonic spectrum $\omega_n = n\omega$ and $n \in \mathbb{Z}$. Therefore, the system can be represented (at a statistical level) by a Hilbert state $q \rightarrow |q\rangle$ defined in the infinite dimensional Hilbert space of complete orthogonal basis $|n\rangle$ associated to the harmonics $q_n(t) \rightarrow |n\rangle$. The time evolution — being cyclic dynamcis — is thus given by the Schrödinger equation $i\partial_t|q\rangle = \hat{H}_{HO}|q\rangle$, where $H_{HO} \rightarrow \hat{H}_{HO}$ defines the Hilbert operator such that $\hat{H}_{HO}|n\rangle = \omega_n|n\rangle$. Similar arguments have been used by 't Hooft for continuous periodic Cellular Automata, e.g. [23–25]. Evidently, this description share deep analogies with the normal ordered Quantum HO (QHO) — for a non relativistic analysis we must assume $n \in \mathbb{N}$.<fnref n="2" /> To establish a possible formal equivalence between the two physics we must however prove that the canonical commutation relations, and in particular the commutation relations among the ladder operators $\hat{a}$ and $\hat{a}^\dagger$, can also be directly inferred from the condition of intrinsic periodicity.

<footnote n="2">² $n \in \mathbb{Z}$ is a consistent description for relativistic physics such as that of the fundamental HOs constituting relativistic fields, as it describes both the positive and negative energy eigenmodes of the field modes.</footnote>
Let us thus impose intrinsic periodicity as constraint by means of PBCs

$$ \hat{q}(t+T) \stackrel{\text{PBCs}}{=} \hat{q}(t), \qquad (2) $$

to the classical solutions $\hat{q}(t)$ and generically describe such an intrinsically periodic HO in a Hilbert space with temporal evolution given by the Schrödinger equation. We prove that the canonical quantization of the HO is formally equivalent to constraining the HO to have of intrinsic periodicity in time. In particular we want to prove, for the specific case of the HO,independently from the previous demonstration of the general results given in previous works [3–12], that

$$ \left\{ \begin{array}{l} \hat{q}(t+T) \stackrel{\text{PBCs}}{=} \hat{q}(t) \\ T = 2\pi/\omega \end{array} \right. \Leftrightarrow [\hat{q}, \hat{p}]_- = i. \qquad (3) $$

The PBCs reproducing this fundamental result must be always understood up to a twist factor, whose only effect is to shift the quantized energy spectrum.

The demonstration starts by assuming, for the parameters $a \to \hat{a}$ and $a^* \to \hat{a}^\dagger$ promoted to generic Hilbert operators and keeping Weyl ordering, the most generic commutation ($-$) or anti-commutation ($+$) relation

$$ [\hat{a}, \hat{a}^\dagger]_\pm = \hat{a}\hat{a}^\dagger \pm \hat{a}^\dagger\hat{a} = \delta, \qquad (4) $$

for some generic constant $\delta \in \mathbb{R}$ (we do not consider here the possibility of a dependency of $\delta$ on the extended phase space coordinates)$^3$.

For any Hilbert operator $\hat{X}$ (or $\hat{Y}$) associated to the intrinsically periodic HO the time evolution prescribed exclusively by the Schrödinger equation is (Heisenberg representation), $\forall t, t' \in \mathbb{R}$,

$$ \hat{X}(t+t') = e^{i\hat{H}_{HO}t'} \hat{X}(t)e^{-i\hat{H}_{HO}t'}. \qquad (5) $$

This can be evaluated by using the Lie series (Baker-Campbell-Hausdorff lemma), see [28],

<footnote n="3">³ A coordinate dependent $\delta$, or more in general a dependency on the extended phase space coordinates, could be relevant for the problem of the quantization of gravity, as manifestation of a minimal length or time at Planck scales. The possibility of such a “generalized uncertainty principle” or similar approaches is investigated in literature, see for instance [26, 27]. As long as we consider free fields ($\omega = \text{const}$) in flat space-time, under the assumption of the ordinary de Broglie-Planck relation, we can however exclude a time dependency of $\delta$. According to our analysis a time dependent $\delta(t)$ in eq.(4) would give $\delta(t) = \delta(t+T) = 1, \forall t$, in eq.(7), so that the commutator must be again $\delta = 1$.</footnote>
in terms of commutation or anti-commutation relations,

$$ 
\begin{aligned}
e^{\hat{X}} \hat{Y} e^{-\hat{X}} &= \hat{Y} + [\hat{X}, \hat{Y}]_{-} + \frac{[\hat{X}, [\hat{X}, \hat{Y}]_{-}]_{-}}{2!} + \dots \\
&= e^{2\hat{X}} \left( \hat{Y} - [\hat{X}, \hat{Y}]_{+} + \frac{[\hat{X}, [\hat{X}, \hat{Y}]_{+}]_{+}}{2!} + \dots \right),
\end{aligned} 
\quad (6) $$

or by using the general identities $e^{\hat{X}} = \sum_n \frac{\hat{X}^n}{n!}$ and $(\hat{X} + \hat{Y})^n = \sum_k \binom{n}{k} \hat{X}^{n-k} \hat{Y}^k$ — notice that the validity of these relations do not depend on commutation relations as assumption.

We write $\hat{a} = a(t)$ and $\hat{a}^\dagger = a^\dagger(t)$, and we study their time evolution. Since $\hat{a}(\hat{a}^\dagger\hat{a})^n = (\hat{a}\hat{a}^\dagger)^n \hat{a}$ and $\hat{a}(\hat{a}\hat{a}^\dagger)^n = (\delta \mp \hat{a}\hat{a}^\dagger)^n \hat{a}$, we have

$$ 
\begin{aligned}
\hat{a}(t+t') &= e^{i\hat{H}_{HO}t'} \hat{a}(t) e^{-i\hat{H}_{HO}t'} = e^{i\hat{H}_{HO}t'} e^{-\frac{i}{2}\omega t'(\hat{a}\hat{a}^\dagger\mp\hat{a}\hat{a}^\dagger+\delta)} \hat{a}(t) \\
&= e^{-\frac{i}{2}\omega t'(\delta\mp\delta)} \hat{a}(t),
\end{aligned} 
\quad (7) $$

and an equivalent relation for its conjugate $\hat{a}^\dagger(t)$.

We must conclude that, $\forall t \in \mathbb{R}$,

$$ 
\left\{ 
\begin{array}{l}
\hat{a}(t+T) \stackrel{\text{PBCs}}{=} \hat{a}(t) \\
T = 2\pi/\omega \Leftrightarrow [\hat{a}, \hat{a}^\dagger]_- = 1.
\end{array}
\right. 
\quad (8) $$

In fact the anti-commutation case $[\hat{a}, \hat{a}^\dagger]_+ = \delta$ as well as the non-commutation case $[\hat{a}, \hat{a}^\dagger]_- = 0$ must be excluded since in both cases we get a trivial evolution $a(t) = \text{const}$, in contraddiction with the Schrödinger evolution — we can also exclude $\delta \in \mathbb{C}$ since its imaginary part would imply a damping in the solutions.

The remaining possibilities compatible with the constraint of intrinsic periodicity are $[\hat{a}, \hat{a}^\dagger]_- = n$ with $n \in \mathbb{N}$. Negative integers $\delta = -n$ imply an inversion of role between the two ladder operators with respect to the corresponding positive integers $\delta = n$.

Finally it is easy to realize that $[\hat{a}, \hat{a}^\dagger]_- = 1$ is the most stringent condition allowed. The quantization obtained with commutator $[\hat{a}, \hat{a}^\dagger]_- = n$ and $n \in \mathbb{N}/\{0, 1\}$ is implicit in the most general quantization $[\hat{a}, \hat{a}^\dagger]_- = 1$. For instance, acting on the vacuum with such a creation operator satisfying commutation equal to generic $n \in \mathbb{N}/\{0, 1\}$ is equivalent to acting $n$ times on the vacuum with the creation operator satisfying $[\hat{a}, \hat{a}^\dagger]_- = 1$. In the $n \in \mathbb{N}/\{0, 1\}$ case however only states $|n\rangle$, $|2n\rangle$, $|3n\rangle \dots$, can be created — where the vacuum state $|0\rangle$ is defined as in usual from, given the assumption eq.(8). This would also mean that the fundamental period of the harmonic system is $T/n$, in contradiction with the hypothesis that the HO has fundamental period $T$.
We may check that, assuming generic commutator $[\hat{a}, \hat{a}^\dagger]_– = \delta$ with $\delta \in \mathbb{R}$, with Weyl-ordered HO Hamiltonian $\hat{H} = \frac{\omega}{2}(\hat{a}^\dagger\hat{a} + \hat{a}\hat{a}^\dagger) = \omega\hat{a}^\dagger\hat{a} + \frac{\omega}{2}\delta$, the Schrödinger evolution implies $\hat{a}(t) = -i\delta\omega\hat{a}$ and thus $\hat{a}(t) = \exp[-i\omega t\delta]\hat{a}(0)$. Imposing fundamental periodicity $\hat{a}(t) = \hat{a}(t+T)$ forces $\delta \in \mathbb{Z}$. Then, the requirement that $T$ is the fundamental periodicity eliminates $|\delta| > 1$, so that the only possibility left is $\delta = 1$.

Our thesis eq.(3) follows directly from eq.(8). The dynamics of an ordinary QHO obtained by imposing canonical commutation relations among the phase-space operators are therefore formally equivalent (at a statistical level) to the dynamics of a corresponding classical HO constrained to have fundamental periodicity $T = 2\pi/\omega$.

The same conclusion holds for the normal ordered Hamiltonian: $H_{HO} := \omega\hat{a}\hat{a}^\dagger$. This shows that the vacuum energy $\frac{1}{2}\omega$ depends on the ordering and not only on the BCs. The PBCs used in this paper must always be interpreted up to a twist factor $e^{-i\theta}$ since both bosonic and fermionic d.o.f typically have $U(1)$ symmetry. For instance the PBCs of the fermionic d.o.f can be replaced with anti-PBCs (twist $e^{-i\pi}$) and the effect on the energy spectrum is a shift of the vacuum energy similarly to that resulting from a different ordering of the operators, [3, 5, 7–12], see also the zitterbewegung in eq.(14). By taking track of the operators ordering, the energy eigenstates and eigenvalues of the QHO can be therefore inferred directly from the condition of intrinsic periodicity, as discussed in the premise of our proof.

## B. Second Quantization of Bosonic Fields

It is now possible to extend our result to the second quantization of a bosonic field. Consider now a classical Klein-Gordon (KG) field

$$ \Phi(\mathbf{x}, t) = \int d^3 p \phi_{\mathbf{p}}(\mathbf{x}, t) = \int d^3 p (f_{\mathbf{p}}(\mathbf{x}, t) \hat{a}_{\mathbf{p}} + f_{\mathbf{p}}^*(\mathbf{x}, t) \hat{a}_{\mathbf{p}}^\dagger), $$

with the corresponding Hamiltonian

$$ H_{KG} = \frac{1}{2} \int d^3 p \omega_{\mathbf{p}} (\hat{a}_{\mathbf{p}} \hat{a}_{\mathbf{p}}^\dagger + \hat{a}_{\mathbf{p}}^\dagger \hat{a}_{\mathbf{p}}). $$

We have infinite d.o.f. but each field component $\phi_{\mathbf{p}}(\mathbf{x}, t)$ can be regarded as an elementary relativistic HO of fundamental periodicity $T_{\mathbf{p}} = 2\pi/\omega_{\mathbf{p}}$, so that the formal equivalence obtained above for the HO can be generalized. The ordinary second quantization of the KG
field is thus formally equivalent to the constraint of intrinsic periodicity

$$ \phi_{\mathbf{p}}(\mathbf{x}, t + T_{\mathbf{p}}) \stackrel{\text{PBCs}}{\equiv} \phi_{\mathbf{p}}(\mathbf{x}, t), \qquad (9) $$

applied for each elementary HO constituting the field. Clearly, $f_{\mathbf{p}}(\mathbf{x}, t)$, $\forall \mathbf{p}$, form a complete set of orthogonal functions due to the PBCs. From eq.(8), $\forall \mathbf{p}, \mathbf{p}_1, \mathbf{p}_2$, we must conclude that

$$ \left\{ \begin{array}{l} \phi_{\mathbf{p}}(\mathbf{x}, t + T_{\mathbf{p}}) \stackrel{\text{PBCs}}{\equiv} \phi_{\mathbf{p}}(\mathbf{x}, t) \\ T_{\mathbf{p}} = 2\pi/\omega_{\mathbf{p}} \end{array} \right. \Leftrightarrow [\hat{a}_{\mathbf{p}_1}, \hat{a}_{\mathbf{p}_2}^{\dagger}]_{-} = \delta^3(\mathbf{p}_1 - \mathbf{p}_2). $$

The KG field $\Phi(\mathbf{x}, t)$ has not a well defined periodicity being the integral of all the momentum components $\phi_{\mathbf{p}}(\mathbf{x}, t)$. Each component has however a well defined periodicity $T_{\mathbf{p}} = 2\pi/\omega_{\mathbf{p}}$. We introduce a *modular* time evolution in which each field component $\phi_{\mathbf{p}}(\mathbf{x}, t)$ has a different time evolution rate $t_{\mathbf{p}}$:

$$ e^{i\hat{\theta}[t]} = e^{\frac{i}{2} \int d^3\mathbf{p} \theta_{\mathbf{p}}[t]} (\hat{a}_{\mathbf{p}}\hat{a}_{\mathbf{p}}^{\dagger} + \hat{a}_{\mathbf{p}}^{\dagger}\hat{a}_{\mathbf{p}}), \qquad (10) $$

with $\theta_{\mathbf{p}}[t] = t_{\mathbf{p}}\omega_{\mathbf{p}}$, $\forall \mathbf{p}$.

We denote by $\Phi(\mathbf{x}, t + [T])$ the KG field in which each component $\phi_{\mathbf{p}}(\mathbf{x}, t)$ has evolved in time exactly of an integer number of corresponding periods $T_{\mathbf{p}}$ with respect to the initial configuration $\Phi(\mathbf{x}, t)$. That is,

$$ \Phi(\mathbf{x}, t + [T]) = e^{i\hat{\theta}[T]} \Phi(\mathbf{x}, t) e^{-i\hat{\theta}[T]}, $$

where $\theta_{\mathbf{p}}[T] = T_{\mathbf{p}}\omega_{\mathbf{p}} = 2\pi$, $\forall \mathbf{p}$.

In terms of such *modular* time evolution we have

$$ \left\{ \begin{array}{l} \Phi(\mathbf{x}, t + [T]) \stackrel{\text{PBCs}}{\equiv} \Phi(\mathbf{x}, t) \\ \theta_{\mathbf{p}}[T] = T_{\mathbf{p}}\omega_{\mathbf{p}} = 2\pi \end{array} \right. \Leftrightarrow [\hat{a}_{\mathbf{p}_1}, \hat{a}_{\mathbf{p}_2}^{\dagger}]_{-} = \delta^3(\mathbf{p}_1 - \mathbf{p}_2). $$

This also means that the second quantization expressed in terms of field commutators holds if and only if each field component is constrained to assume the same configuration after an integer number of its natural cycles

$$ \left\{ \begin{array}{l} \Phi(\mathbf{x}, t + [T]) \stackrel{\text{PBCs}}{\equiv} \Phi(\mathbf{x}, t) \\ \theta_{\mathbf{p}}[T] = T_{\mathbf{p}}\omega_{\mathbf{p}} = 2\pi, \forall \mathbf{p} \end{array} \right. \Leftrightarrow [\Phi(\mathbf{x}, t), \dot{\Phi}(\mathbf{y}, t)]_{-} = i\delta^3(\mathbf{x} - \mathbf{y}). $$

Summarizing, a *classical* KG field $\Phi(\mathbf{x}, t)$ is formally equivalent to a *second quantized* KG field if and only if each component $\phi_{\mathbf{p}}(\mathbf{x}, t)$ is constrained to be intrinsically periodic
with respect to its natural recurrence. In a similar way we can infer the formal equivalence between second quantization and the constraint of intrinsic periodicity for vectorial fields, with all the due prescriptions aside the ordinary second quantization of gauge theories, see also sec.(IV) and [3, 5].

### III. SECOND QUANTIZATION OF FERMIONS

#### A. Fermi and Dirac oscillator

The analysis of the Fermi oscillator will allow us to easily generalize the above formal equivalence to Dirac fields. Assuming fundamental periodicity $T = 2\pi/\omega$, the Fermi oscillator is given by the following Hamiltonian

$$ \hat{H}_F = \omega \hat{f}^\dagger \hat{f}. $$

We want to prove that the condition of fundamental periodicity — up to a global twist factor — imposed as constraint

$$ \hat{f}(t+T) \stackrel{\text{PBCs}}{=} \hat{f}(t) \qquad (11) $$

necessarily implies the canonical anti-commutators of fermionic d.o.f.. The implicit twist factor $e^{-i\theta}$ in the PBCs means that the results proven below are also compatible with anti-PBCs (Neveu-Schwarz), i.e. $\theta = \pi$. From a physical point of view (spin-statistics) anti-PBCs is the most appropriate BCs to assume for our geometric derivation of second quantization for fermions. However, for the sake of simplicity, we will use PBCs for our proofs, bearing in mind that this are defined up to a twist factor, whose only effect is the introduction of a vacuum energy (shift of the energy spectrum).

Again, the condition of intrinsic periodicity (or, anti-periodicity) guarantees that the system can be described, at a statistical level, in a Hilbert space and that its temporal evolution is given by the Schrödinger equation. Thus we promote $\hat{f}$ and $\hat{f}^\dagger$ to Hilbert operators and we assume generic anti-commutation

$$ [\hat{f}, \hat{f}^\dagger]_+ = \delta, \qquad (12) $$

with $\delta \in \mathbb{R}$.

We want to obtain the Fermi-Dirac statistics so we directly exclude the possibility of commutation relations $[\hat{f}, \hat{f}^\dagger]_- = \delta$, and we assume $\hat{f}^2 = 0$ for obvious reasons. Without
loss of generality we temporarily assume $\omega = 1$. Since $\hat{H}_F^2 = \delta\hat{H}_F$, the eigenvalues of $\hat{H}_F$ must be $\lambda^2 = \delta\lambda$, so that $\lambda = 0$ or $\lambda = \delta$. Let us investigate the quantization resulting from the generic anti-commutation relation eq.(12). We define $|0\rangle$ such that $\hat{H}_F|0\rangle = 0$ with $\langle 0|0\rangle = 1$. Thus $|a\rangle = \hat{f}|0\rangle = 0$ since $|a\rangle|^2 = \langle 0|\hat{f}^\dagger\hat{f}|0\rangle = 0$. Also we define $|\delta\rangle = \hat{f}^\dagger|0\rangle$ such that $|\delta\rangle|^2 = \langle 0|(\delta - \hat{f}^\dagger\hat{f})|0\rangle = \delta$.

In analogy with the bosonic case, from eq.(5) and eq.(6) we find the following time evolution

$$ \hat{f}(t + t') = e^{-i\omega t'(\delta - 2\hat{f}^\dagger\hat{f})} \hat{f}(t) = e^{i2\hat{H}_F t'} e^{-i\omega t\delta} \hat{f}(t). $$

Notice the *Zitterbewegung* term in this single-particle description, e.g. [29].

From the condition of intrinsic periodicity $\hat{f}(t+T) \stackrel{\text{PBCs}}{=} \hat{f}(t)$ we immediately have $\delta \in \mathbb{N}$. In fact, $\hat{H}_F|\delta\rangle = \hat{f}^\dagger\hat{f}\hat{f}^\dagger|0\rangle = \hat{f}^\dagger(\delta - \hat{f}^\dagger\hat{f})|0\rangle = \hat{f}^\dagger\delta|0\rangle = \delta|\delta\rangle$. Thus $(\delta - 2\delta) \in \mathbb{Z}$ and finally $\delta \in \mathbb{Z}$. Similarly, $\hat{H}_F|0\rangle = 0$ and again $\delta \in \mathbb{Z}$. The choice $-\delta \in \mathbb{N}$ is physically equivalent to $\delta \in \mathbb{N}$ for the same reason given in the bosonic case.

The case $\delta = 0$ is trivial and must be excluded because it yields $\hat{f}(t) = \text{const}$. The case $\delta = 1$ works fine. In particular $\hat{f}(t+t')|1\rangle = \exp(-i\omega t')\hat{f}(t)|1\rangle$ and $\hat{f}^\dagger(t+t')|0\rangle = \exp(+i\omega t')\hat{f}^\dagger(t)|0\rangle$. The case $\delta = n$ with $n \in \mathbb{N}/\{0, 1\}$ must be excluded as well. It implies $\hat{f}(t+t')|n\rangle = \exp(-in\omega t')\hat{f}(t)|n\rangle$ and $\hat{f}^\dagger(t+t')|0\rangle = \exp(+in\omega t')\hat{f}^\dagger(t)|0\rangle$. However $|n\rangle$ cannot be interpreted as $n$ fermions in the same state due to the underlying Dirac-Fermi statistics. It must be rather interpreted as a fermion of fundamental energy $n\omega$ in its state $|1\rangle$ since no lower energetic states can be created. In other words, similarly to the bosonic case, it means that the period $T$ of the Hamiltonian is not the fundamental period of the system. The fundamental period in this case would be $T_n = T/n$ in contradiction with our hypothesis.

This can be double-checked also by considering that the generic anti-commutator eq.(12) plus the condition $\hat{f}^2 = 0$ yields, through the Schrödinger evolution implicit in the intrinsically periodic dynamics, $\hat{f}(t) = \exp[-i\delta\omega t]\hat{f}(0)$ and thus $\delta = 1$ from the requirement of fundamental periodicity $T$ and non-trivial evolution.

The only possible choice in eq.(12) is $\delta = 1$. Thus,

$$ \left\{ \begin{array}{l} \hat{f}(t+T) \stackrel{\text{PBCs}}{=} \hat{f}(t) \\ T = 2\pi/\omega \end{array} \right. \Leftrightarrow [\hat{f}, \hat{f}^\dagger]_+ = 1. $$
The same conclusion also apply to

$$ \hat{H}'_F = \frac{1}{2}\omega(\hat{f}^\dagger\hat{f} - \hat{f}\hat{f}^\dagger), \qquad (13) $$

where the time evolution is now a pure Zitterbewegung,

$$ \hat{f}(t + t') = e^{-i\omega t'(\delta - 2\hat{f}^\dagger\hat{f})} \hat{f}(t) = e^{i2\hat{H}'_F t'} \hat{f}(t), \qquad (14) $$

with the same Fock space of the previous case. This shows that the dynamics generated by $\hat{H}_F$ is equivalent, up to a constant shift of the energy $\omega/2$, to the pure “zitterbewegung” dynamics generated by $\hat{H}'_F$. In other words, second quantization for fermionic fields is equivalently obtained by imposing anti-PBCs rather than PBCs. All the relevant results of this paper are unaffected by whether we assume PBCs, anti-PBCs, or more in general twisted PBCs.

In analogy to the Fermi oscillator we may consider the Dirac oscillator defined by the Hamiltonian

$$ \hat{H}_D = \omega(\hat{b}^\dagger\hat{b} - \hat{d}\hat{d}^\dagger). $$

Similarly to eq.(13), the generic anti-commutation relations eq.(12) assumed separately for both $\hat{b}$ and $\hat{d}$ operators lead to the evolutions

$$ \hat{b}(t + t') = e^{-i\omega t'(\delta - 2\hat{b}^\dagger\hat{b})} \hat{b}(t), \quad \hat{d}(t + t') = e^{-i\omega t'(\delta - 2\hat{d}^\dagger\hat{d})} \hat{d}(t). $$

The Zitterbewegung is disappeared in the Dirac oscillator and thus it will disappear in the Dirac field description, as known in literature, e.g. [29, 30].

By introducing the ordinary concept of Dirac vacuum and in analogy with the Fermi oscillator we obtain

$$ \left\{ \begin{array}{l} \hat{b}(t+T) \stackrel{\text{PBCs}}{\equiv} \hat{b}(t) \\ \hat{d}(t+T) \stackrel{\text{PBCs}}{\equiv} \hat{d}(t) \\ T = 2\pi/\omega \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} [\hat{b}, \hat{b}^\dagger]_+ = 1 \\ [\hat{d}, \hat{d}^\dagger]_+ = 1, \end{array} \right. $$

and all the other possible anti-commutators must be equal to zero.

## B. Second Quantization for Dirac fields

We want to establish the relationship between PBCs and the second quantization of Dirac fields in analogy to the previous analysis of bosonic fields.
The Dirac field can be written in the following form

$$ \Psi(\mathbf{x}, t) = \int d^3p \psi_{\mathbf{p}}(\mathbf{x}, t) = \int d^3p (u_{\mathbf{p}}(\mathbf{x}, t)\hat{b}_{\mathbf{p}} + v_{\mathbf{p}}(\mathbf{x}, t)\hat{d}_{\mathbf{p}}^{\dagger}), $$

where $u_{\mathbf{p}}(\mathbf{x}, t)$ and $v_{\mathbf{p}}(\mathbf{x}, t)$ are spinors (we suppress the spin index). The Hamiltonian is

$$ H_{\Psi} = \int d^3p \omega_{\mathbf{p}} (\hat{b}_{\mathbf{p}}^{\dagger}\hat{b}_{\mathbf{p}} - \hat{d}_{\mathbf{p}}\hat{d}_{\mathbf{p}}^{\dagger}). $$

By investigating the time evolution of these ladder operators, in complete analogy with the Dirac HO, $\forall \mathbf{p}, \mathbf{p}_1, \mathbf{p}_2$ we obtain the following formal equivalence,

$$ \left\{ \begin{array}{l} [\hat{b}_{\mathbf{p}_1}, \hat{b}_{\mathbf{p}_2}^{\dagger}]_{+} = \delta^3(\mathbf{p}_1 - \mathbf{p}_2) \\ [\hat{d}_{\mathbf{p}_1}, \hat{d}_{\mathbf{p}_2}^{\dagger}]_{+} = \delta^3(\mathbf{p}_1 - \mathbf{p}_2) \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} \psi_{\mathbf{p}}(\mathbf{x}, t + T_{\mathbf{p}}) \stackrel{\text{PBCs}}{\equiv} \psi_{\mathbf{p}}(\mathbf{x}, t) \\ T_{\mathbf{p}} = 2\pi/\omega_{\mathbf{p}}, \end{array} \right. $$

and all the other anti-commutators equal to zero.

In terms of the *modular* time evolution analogous to eq.(10) the above result can be written as

$$ \left\{ \begin{array}{l} \Psi(\mathbf{x}, t + [T]) \stackrel{\text{PBCs}}{\equiv} \Psi(\mathbf{x}, t) \\ \theta_{\mathbf{p}}[T] = T_{\mathbf{p}}\omega_{\mathbf{p}} = 2\pi \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} [\hat{b}_{\mathbf{p}_1}, \hat{b}_{\mathbf{p}_2}^{\dagger}]_{+} = \delta^3(\mathbf{p}_1 - \mathbf{p}_2) \\ [\hat{d}_{\mathbf{p}_1}, \hat{d}_{\mathbf{p}_2}^{\dagger}]_{+} = \delta^3(\mathbf{p}_1 - \mathbf{p}_2), \end{array} \right. $$

and finally

$$ \left\{ \begin{array}{l} \Psi(\mathbf{x}, t + [T]) \stackrel{\text{PBCs}}{\equiv} \Psi(\mathbf{x}, t) \\ \theta_{\mathbf{p}}[T] = T_{\mathbf{p}}\omega_{\mathbf{p}} = 2\pi, \forall \mathbf{p} \end{array} \right. \Leftrightarrow [i\Psi^{\dagger}(\mathbf{x}, t), \Psi(\mathbf{y}, t)]_{+} = i\delta^3(\mathbf{x} - \mathbf{y}). $$

The condition of intrinsic periodicity is formally equivalent to the second quantization condition also for Dirac fields.

## IV. COMMENTS ABOUT INTERACTION

Even though it was sufficient to restrict our analysis to free fields to prove second quantization, interactions can be then included in several, conceptually distinct, ways. Second quantized fields is in fact the natural framework in which interactions are then introduced to study scattering amplitudes.

As a direct consequence of the equivalence between intrinsic periodicity and the canonical commutation relations of second quantization, the PBCs imposed above, of eq.(9), can be regarded as a purely mathematical device to generate the canonical (anti-)commutators of
standard QFT (without postulating them), together with the corresponding ladder-operator
algebra of ordinary QFT. Once the canonical algebra is obtained, one may the safely proceed
exactly as in standard QFT, without explicitly tracking if second quantization is obtained
from the underlying intrinsic periodicity or it is postulated as usual. Interactions can then
be introduced in the familiar way: given the commutation relations, quantum observables
follow from the standard perturbative expansion in Fock space and from the scattering
matrix.

Nevertheless the compact support offered by the PBCs could result in useful mathe-
matical tools, for instance, in regulating integrals in Feynman diagrams or in improving
calculation techniques of quantum amplitudes. This is also suggested by the analogy with
finite-temperature QFT, where the “mathematical trick” of compactifying Euclidean time re-
veals a relation between quantum and classical (statistical) dynamics, closely paralleling the
compactification of Minkowskian time considered here [31]. In the Matsubara formalism,
quantization at temperature $T$ amounts to imposing (anti-)periodic boundary conditions
with period $\beta = 1/(k_B T)$ in Euclidean time [19]; in our framework, by contrast, the PBCs
are mode-dependent and are imposed via the modular time operator. This analogy sug-
gests that topological techniques akin to those used in finite-temperature QFT and in other
non-trivial topological models may provide new insight into the renormalization of Feynman
diagrams in ordinary QFT [32].

A. General Covariance Method and Geometrodynamical Description of Gauge in-
teraction

The most general prospective to include interaction in this work is to treat the “inter-
nal time coordinates” of intrinsically cyclic nature as a fundamental ingredient underlying
particle’s dynamics. Within ECT, this viewpoint leads to a geometrodynamical description
of interactions, developed in [5] and shortly reviewed below — the exact geometrodynamics
accounting for the electromagnetic interaction was obtained in [37], after the publication of
the present manuscript.

To set the analysis in a covariant way, note that the intrinsic temporal periodicity con-
dition in eq.(9) for a KG mode implies, through the free Equations of Motion (EoM), an
associated spacetime recurrence of the corresponding classical solution. For a free mode with
four-momentum $p_μ = \{\omega, -\mathbf{p}\}$, the field component can be written as $\phi_{\mathbf{p}}(x) \propto \exp[-ip_μx^μ]$ so that the periodicity conditions can be equivalently expressed as a four-dimensional recurrence

$$ \phi_{\mathbf{p}}(x^{\mu}) \stackrel{\text{PBCs}}{=} \phi_{\mathbf{p}}(x^{\mu} + \Lambda_{\mathbf{p}}^{\mu}). \qquad (15) $$

where the spacetime recurrence $\Lambda_{\mathbf{p}}^{\mu} = \{T_{\mathbf{p}}, \lambda_{\mathbf{p}}\}$ is a contravariant four-vector such that it co-modulate with $p_μ$ forming the invariant $p_μ\Lambda_{\mathbf{p}}^{\mu} = 2\pi$. In other words $T_{\mathbf{p}} = 2\pi/\omega$ and $\lambda_{\mathbf{p}}$ is the ordinary de Broglie wavelength vector ($\lambda^i p_i = 2\pi, i = 1, 2, 3$). In the free case, $p_μ$ is constant and therefore the recurrence $\Lambda_{\mathbf{p}}^{\mu}$ is global.

Using this spacetime recurrence, a single KG mode can be isolated by imposing PBCs directly at the level of the action,

$$ S_{\text{Free}} = \oint_{X^{\mu}}^{X^{\mu} + \Lambda_{\mathbf{p}}^{\mu}} d^4 x \mathcal{L} (\partial_{\mu} \phi_{\mathbf{p}}(x), \phi_{\mathbf{p}}(x)), \qquad (16) $$

where $\mathcal{L}$ is the free KG Lagrangian density. The notation $\oint_{X^{\mu}}^{X^{\mu} + \Lambda_{\mathbf{p}}^{\mu}}$ indicates that the PBCs eq.(15) are assumed to select the local solution of corresponding spacetime recurrence $\Lambda_{\mathbf{p}}^{\mu}$ in each point $x = X$. The PBCs also means that the corresponding intrinsically periodic solution $\phi_{\mathbf{p}}(x)$ is then automatically second quantized, in the sense established by the main result of this paper. Eq.(16) therefore provides the action-level counterpart of the intrinsically periodic harmonic degree of freedom used above to discuss second quantization.

Interactions can now be introduced as local modulations of the spacetime recurrence, induced by local transformation of the underlying Minkowski coordinates (in the linearized regime adopted here), which deforms the free harmonic dynamics into a locally anharmonic one. Switching on an interaction amounts to replacing the constant free four-momentum $p_μ$ with a locally varying four-momentum $p'_μ(x)$. Since $p'_μ(x)$ locally determines the phase evolution through the de Broglie-Planck relation, the global recurrence $\lambda_{\mathbf{p}}^{\mu}$ must be replaced by a corresponding local contravariant four-vector $\Lambda_{\mathbf{p}}^{\mu}(x)$. In this picture, the compact spacetime domain in eq.(16) is locally “deformed”, i.e. the compactification lengths are modulated in spacetime. In analogy with GR (see Ref. [5] for details), where gravitational interaction is directly linked to clock redshift and rulers contraction, this is obtained by locally deforming the flat spacetime of the free theory.

In order to introduce interactions we thus allow for a generic local deformation of the spacetime frame, encoded by a vierbein field $e^a_μ(x)$ (with inverse $e^a_μ(x) = \delta^a_μ$).

$$ e^a_μ(x) e^a_ν(x) = \delta^a_ν, \qquad e^a_μ(x) e^a_ν(x) = \delta^a_μ. $$
Equivalently, at a generic interaction point $x = X$ one may write the local relation between differentials as

$$dx^{\mu} \xrightarrow{\text{interaction}} dx'^{\mu}(X) = e^{\mu}_{a}(X) dx^{a}, \qquad (17)$$

where $x^a$ denote locally inertial coordinates (tangent-space indices are raised/lowered with $\eta_{ab}$).

Consistency with the corresponding *local* deformation of the compact spacetime boundary implies a *local* modulation of the spacetime recurrence. In particular, the free constant recurrence $\lambda_{\mathbf{p}}^{\mu}$ is replaced by a locally varying one, which we can represent by the integrated recurrence

$$\Lambda_{\mathbf{p}}^{\mu} \xrightarrow{\text{interaction}} \Lambda_{\mathbf{p}'}^{\mu}(X) \qquad (18)$$

i.e. the image of the free cycle under the local frame field.<fnref n="4" />

In turn, the local modulation of the recurrence implies a *local* modulation of the phase of the field solution, i.e. a local variation of four-momentum. In tetrad form this is expressed as

$$p_{\mu} \xrightarrow{\text{interaction}} p'_{\mu}(X) = e^{a}_{\mu}(X) p_{a}, \qquad (19)$$

as required to describe interactions.

Equivalently, we may say that the interaction is introduced by locally deforming the flat Minkowski metric $\eta_{\mu\nu}$ into an effective deformed metric (not necessarily curved, as we will see),

$$\eta_{\mu\nu} \xrightarrow{\text{interaction}} g_{\mu\nu}(x) = e^{a}_{\mu}(x) e^{b}_{\nu}(x) \eta_{ab}. \qquad (20)$$

Under this local transformation the free action in eq.(16) is mapped into an action with locally varying boundary, eq.(23) below. The recurrence rates vary locally (and causally) together with the locally varying four-momentum during the interaction, showing that relativistic locality and causality are preserved despite the underlying cyclic structure introduced in eq.(16). Notice also that the interaction can be encoded in the dynamics of the boundary, in a kind of holographic description [5, 6].

$$\lambda_{\mathbf{p}'}^{\mu}(x) := e^{\mu}_{a}(x) \Lambda_{\mathbf{p}}^{a},$$

from the *integrated recurrence* $\Lambda_{\mathbf{p}'}^{\mu}(X)$, which appears in the PBCs and transforms as a coordinate displacement. In general, $\lambda_{\mathbf{p}}^{\mu} \neq \Lambda_{\mathbf{p}'}^{\mu}$. By construction one has the local phase-harmony condition $p'_{\mu}(x) \lambda_{\mathbf{p}'}^{\mu}(x) \equiv 2\pi$ for all $x$. In the weak-field (slowly varying) limit, $\lambda_{\mathbf{p}'}^{\mu}(x)$ can be identified with $\Lambda_{\mathbf{p}'}^{\mu}(x)$ to leading order, and the free case is recovered as $\lambda_{\mathbf{p}}^{\mu} \equiv \Lambda_{\mathbf{p}}^{\mu}$ [5].

<footnote n="4">⁴ It is useful to distinguish the *local* (tangent) recurrence rate (a contravariant spacetime vector transforming as $dx^{\mu}$)</footnote>
Two types of spacetime geometrodynamics are then possible: one reproduces gravita-
tional interaction as in standard GR; the other-enabled by the non-trivial topology of the
compactified description [5–7] — is naturally identified with ordinary gauge interaction.

Two types of spacetime geometrodynamics are then possible: one reproduces gravitational interaction as in standard GR and is obtained by local diffeomorphisms yielding curved spacetime; the other — enabled by the non-trivial topology of the compactified description [5–7] — is naturally identified with ordinary gauge interaction and is obtained by local isometries distorting spacetime.

1. *Gravitational Interaction*

In the first case we consider the standard class of local diffeomorphisms that induce
an effective *curved* spacetime metric $g_{\mu\nu}(x)$. This corresponds to gravitational interaction
in the usual sense of GR. In particular, it reproduces the familiar modulation of clock
rates (gravitational redshift) and the corresponding ruler contractions/expansions encoded,
for example, in the Schwarzschild geometry and in other textbook GR settings [36]. For
example, in the linear approximation, gravitational interaction can be consistently derived
in these terms — thus Einstein's equation recovered by requiring general invariance [36].
Because the dynamics is formulated with PBCs, the locally modulated solutions of the
action obtained from eq.(16) under such transformations — see also eq.(23) below — are
implicitly second quantized according to the main result of this paper. This suggests that the
present framework may provide a useful route for discussing the quantization of gravitational
degrees of freedom, at least at the level where the interaction is captured by local spacetime
modulations.

2. *Gauge Interaction*

The second type of transformations reveals an inedited geometrodynamical origin of gauge
interaction, as reported in [5-7] (see also original Weyl's proposal). In particular we will
investigate the geometrodynamical origin of the electromagnetic gauge interaction. This is
exclusively possible because the condition of intrinsic periodicity, written in terms of PBCs
on a compact space-time, introduces a non-trivial topology. Any naturally occurring vector
bundle, such as the tangent bundle, and other possible ones, must now come equipped with connections when anything related to them is varied. The resulting gauge invariance in the theory plays a central role in the proof of the equivalence between intrinsically periodic dynamics and canonical quantization for general Hamiltonian systems [3].

To obtain gauge interaction let us consider local isometries of spacetime, such as local Lorentz transformations. in order to reproduce electromagnetism we assume that the isometry is an abelian $U(1)$. The only effect of this local isometries are a local “rotations” of the action space-time boundary while the metric is kept locally flat<fnref n="5" />

$$ \sqrt{-g(X)} \approx 1. $$

We know that these kind of transformations have no effects on ordinary QFT, where gauge interaction is in fact introduced by postulating gauge invariance — due to its trivial topology. However, due to the compact nature of space-time in ECT, such local “rotations” of the boundary clearly imply, through the PBCs, local variations of space-time recurrence $\Lambda_{\mathbf{p}}^{\mu} \to \Lambda_{\mathbf{p}'}^{\mu}(X)$ in the locally modulated solution $\phi_{\mathbf{p}}(X) \to \phi_{\mathbf{p}'}(X)$ of the transformed action, see eq.(23) below. In turns this implies corresponding local variations of four-momentum $p_{\mu} \to p'_{\mu}(x)$ in the solution, that is to say interaction, despite the flat metric. In [5], we have proven that, in the case of local isometries “rotating” the space-time boundary, this yields ordinary gauge interaction (without postulating gauge invariance).

To verify this, let us consider the simple case of a local $U(1)$ “rotation” of space-time boundary in eq.(16), as induced by the following local infinitesimal isometry,

$$ e_{\mu}^{a}(x) = \delta_{\mu}^{a} - e \xi_{\mu}^{a}(x), \qquad (21) $$

where $\xi_{\mu}^{a}(x)$ is associated to peculiar Killing vector fields of a $U(1)$ isometry, see eq.(23). The resulting local variation of four-momentum in the solution can be written in the form of a minimal substitution

$$ p_{\mu} \xrightarrow{U(1) \text{ isometry}} p'_{\mu}(x) = p_{\mu} - e A_{\mu}(x), \qquad (22) $$

where $e$ and

$$ A_{\mu}(x) = \xi_{\mu}^{a}(x) p_{a}, $$

<footnote n="5">⁵ For the sake of simplicity, here we neglect the possibility of a local scaling factor, [5–7].</footnote>
will be in a moment identified with the electric charge and the ordinary gauge field, respectively.

Let us investigate more in detail how the solution of the action eq.(16) is transformed under this isometry. For instance we may consider the transformation of a single fundamental frequency mode allowed by the PBCs. According to the result of this paper, the fundamental mode coincides with the single field mode of a KG field selected by the PBCs — the higher vibrational modes are the quantum excitations of the particle. The fundamental mode solution of eq.(16) is

$$ \phi_{\mathbf{p}}(x) = e^{-ip_{\mu}x^{\mu}} \phi_{\mathbf{p}}(0). $$

Under the local $U(1)$ isometry, the transformed action results in transformed boundaries while the Lagrangian density is not affected by the transformation

$$ S_{\text{Free}} \xrightarrow{U(1) \text{ isometry}} S_{\text{EM}} = \oint_{X^{\mu}}^{X^{\mu} + \Lambda'_{\mu}(X)} d^4 x \mathcal{L} (\partial_{\mu} \phi'_{\mathbf{p}'}(x), \phi'_{\mathbf{p}'}(x)). \quad (23) $$

As a consequence of the transformed boundary of the theory the fundamental solution of the transformed action is a locally modulated wave

$$ \phi_{\mathbf{p}}(X) \xrightarrow{U(1) \text{ isometry}} \phi'_{\mathbf{p}'}(X) = e^{-i \int_{0}^{X^{\mu}} p'_{\mu}(x) dx^{\mu}} \phi_{\mathbf{p}}(0) \\ = V(X) \phi_{\mathbf{p}}(X), $$

where the parallel transport,

$$ V(X) = e^{i e \int_{0}^{X^{\mu}} \Lambda_{\mu}(x) dx^{\mu}}, $$

describing the internal transformation under the isometry, is formally a path dependent Wilson line of a gauge connection.

It can be used to equivalently write the locally modulated solution $\phi'_{\mathbf{p}'}(x)$ of the action of locally varying compactification four-length $\Lambda'_{\mu}(x)$, see eq.(23), as the solution of the original free action $S_{\text{Free}}$ of globally fixed compactification four-length $\Lambda_{\mu}$ and gauged KG Lagrangian density. In fact, we can substitute,

$$ \phi_{\mathbf{p}}(x) = V^{-1}(x) \phi'_{\mathbf{p}}(x), $$

directly into the action eq.(16), so that the transformed action eq.(23) can be equivalently rewritten in the following, more familiar form

$$ \begin{aligned} S_{\text{Free}} \rightarrow S_{\text{EM}} &= \oint_{X^{\mu}}^{X^{\mu} + \Lambda_{\mu}^{\mu}} d^4 x \mathcal{L} (V^{-1}(x) D_{\mu} \phi'_{\mathbf{p}'}(x), V^{-1}(x) \phi'_{\mathbf{p}'}(x)) \\ &= \oint_{X^{\mu}}^{X^{\mu} + \Lambda_{\mu}^{\mu}} d^4 x \mathcal{L} (D_{\mu} \phi'_{\mathbf{p}'}(x), \phi'_{\mathbf{p}'}(x)), \end{aligned} \quad (24) $$
where the covariant derivative is

$$D_{\mu} = \partial_{\mu} - ieA_{\mu}(x).$$

In other words, the locally transformed action eq(23) is equivalent to the gauged free action $S_{Free}$, having the same solution $\phi'_{\mathbf{p}'}(x)$. The covariant derivative tune the locally modulated periodicity $\Lambda'_{\mathbf{p}'}(X)$ of $\phi'_{\mathbf{p}'}(x)$ with the PBCs a fixed boundary $\Lambda_{\mathbf{p}}^{\mu}$ imposed by the action.

The fixed PBCs of the action eq.(25) implies that the possible kinetic term $A_{\mu}$ must be gauge invariant, so that recurrences of $A_{\mu}$ can be tuned (adjusted) by parallel transport and covariant derivatives to the recurrence imposed by the action boundary — besides requiring locality and Lorentz invariance, [5]. Essentially, this is precisely the standard justification that leads to the Maxwell kinetic term in ordinary field theory — in par.(??) will give a further, purely 4D geometric justification in terms of KK mechanism. The unique lowest- dimension term allowed is therefore of the form $-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$. Therefore we get the standard EM action as the action describing the local $U(1)$ isometries of spacetime

$$S_{Free} \rightarrow S_{EM} = \oint_{X^{\mu}}^{X^{\mu} + \Lambda_{\mathbf{p}}^{\mu}} d^{4}x \left[ -\frac{1}{4} F^{\mu\nu} F_{\mu\nu} + \mathcal{L} (D_{\mu} \phi'_{\mathbf{p}'}(x), \phi'_{\mathbf{p}'}(x)) \right], \quad (25)$$

Without postulating gauge invariance we have obtained the gauged KG theory as a consequence of local $U(1)$ isometries of the (topologically non-trivial) space-time coordinates of ECT, in agreement with the *Principle of General Covariance* (extended to gauge interaction, i.e. theory with non-trivial space-time geometry). See [5] for more details about the correspondence with ordinary gauge interaction such as gauge invariance, the kinematic term of the gauge field and the Noether currents.

In the subsequent paper [1] we have precisely established that the local isometries responsible for the EM gauge are Lorentz boosts which, described in terms of Fermi-Walker transport actually leads to Maxwell law and Lorentz force. Spatial rotations are instead holonomies that contribute as pure gauge transformation defining gauge orbits. Furthermore we have been able to derive the Maxwell kinetic term from a purely 4D reinterpretation of the Kaluza-Klein mechanism, allowed aging by the non trivial topology of the theory which associates a fiber $S^1$ (particle's internal clock) to the particle's dynamics.

Since the global PBCs of the above actions implicitly second quantize the solution, we expect that the theory automatically describes (bosonic) QED. Actually, in [5] we have also
proven that the intrinsically periodic solution of the above action is formally solution of
the ordinary Feynman path integral of (bosonic) QED, in full confirmation of the general
correspondence between intrinsically cyclic dynamics and quantum dynamics investigated
in this paper and in the previous ones [3–12].

V. CONCLUSIONS

The constraint of intrinsic time periodicity can be formally used in place of the canonical
commutators in the second quantization for both bosonic and fermionic fields — in analogy
with the “quantization” of the mass spectrum in the Kaluza-Klein theory where the intrinsic
periodicity is on the cyclic extra-dimension. This study lays the ground of a new mathemat-
ical method potentially applicable in calculating all quantum observables in QFT, such as in
Feynman diagrams, and is essentially based on the physics of classical-relativistic harmonic
systems. The result can also be relevant to address the problem of time in the quantization
of gravity, on the line of Rovelli’s proposal [22].

Even though the constraint of intrinsic periodicity can be safely used as a simple “math-
ematical trick” to quantize fields, we believe that its origin has strong physical motivations
in a renewed formulation of the concept of relativistic time in terms of “internal times” of
intrinsically cyclic nature.

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