Paper VII — Variable Particle Number and Field-Theoretic Extension

DOI: 10.5281/zenodo.17925704 Variable Particle Number and Field-Theoretic Extension Paper VII of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Contents 1 Introduction and Logical Position 1 2 Local Additive Independence of Excitations 2 3 Bounded Local Information Density 2 4 Fock Space as the Unique Resolution 3 5 Fields as Operator-Valued Distributions 3 6 Canonical (Anti)Commutation Relations 3 7 Vacuum, Normal Ordering, and Operational Grounding 3 8 Conclusion 4 DOI: 10.5281/zenodo.17925704 apers I–VI reconstructed quantum kinematics, relativistic dynamics, interactions, spin, statistics, and many-body entanglement from bounded information capacity and operational locality. The present paper completes the transition from finite-particle quantum mechanics to quantum field theory. We introduce two operational principles: Local Additive Independence of Excitations (LAIE) and Bounded Local Information Density (BLID). We prove that these jointly force variable particle number, Fock space structure, and operator-valued distributions as the unique mathematical realization of local observables. Fields are shown not to be fundamental objects but necessary local information buffers reconciling infinite global capacity with finite operational resolution. Canonical (anti)commutation relations, normal ordering, and the restriction to inertial representations emerge as operational necessities. This reconstructs free relativistic quantum field theory without postulating fields or quantization rules. Contents 1 Introduction and Logical Position Papers I–VI established that quantum theory is the unique dynamical realization of finite information capacity under locality and causal order. The remaining structural transition is the passage from fixed-particle-number systems to quantum fields. Standard presentations of quantum field theory proceed by postulating classical fields and imposing canonical quantization rules. This paper reverses the logic. Core question. What operational problem cannot be solved within finite-particle Hilbert spaces, and what structure is forced when that problem is posed? – 1 – Answer. If arbitrarily many indistinguishable excitations can be created locally, while each finite region admits only finite operational resolution, then: •Particle number cannot be fixed (forcing Fock space), •Local observables cannot be operators at points (forcing distributions), •The only consistent bookkeeping structure is a quantum field. Fields are not added—they are forced. 2 Local Additive Independence of Excitations [Local Additive Independence of Excitations (LAIE)] For any set of disjoint spacelike-separated regions R1, . . . , Rn, there exist operational procedures allowing the independent preparation of excitations in each region such that: 1. Preparations in distinct regions commute operationally, 2. The joint preparation does not alter the intrinsic properties of individual excitations. Remark 1.LAIE is the many-body extension of Operational Independence (Paper VI). It expresses the ability to populate the universe with excitations without global coordination. LAIE implies that no finite upper bound exists on the global excitation number. Proof. Suppose a global bound Nexists. One may choose N+ 1 spacelike-separated regions and apply LAIE to prepare one excitation in each. The inability to do so would imply nonlocal interference (action at a distance) preventing a local operation. Thus, global capacity must be unbounded. 3 Bounded Local Information Density [Bounded Local Information Density (BLID)] For any bounded spacetime region Rand finite operational resolution scale ϵ, there exists a finite upper bound on the distinguishable information content accessible within R. Remark 2.BLID is the continuum analogue of bounded information capacity (Paper I). It excludes the possibility of infinite-resolution measurements (singularities) in finite regions. LAIE and BLID are jointly incompatible with fixed-particle-number Hilbert spaces. Proof. Fixed-NHilbert spaces (N) scale polynomially. LAIE demands an exponential growth in capacity as space is partitioned (Fock structure). Conversely, allowing arbitrarily many local degrees of freedom within a finite region violates BLID. Thus, a new structure must mediate the tension between infinite global capacity and finite local limits. – 2 – 4 Fock Space as the Unique Resolution [Fock space necessity] The unique Hilbert-space structure compatible with LAIE and BLID is a bosonic or fermionic Fock space: F±(1) = ∞ M n=0 ⊗±n 1. Proof. LAIE requires additive composition of excitation numbers across regions. BLID requires that locally, the state space resembles a finite collection of oscillators. The only associative mathematical construction that supports variable particle number, indistinguishability (from Paper V), and locality is the Fock sum of symmetric or antisymmetric tensor powers. 5 Fields as Operator-Valued Distributions [Field necessity] Local observables compatible with LAIE and BLID must be operator-valued distributions, not functions. Proof. Suppose local observables were defined at points, ϕ(x). To measure a field value at a point with perfect precision implies infinite uncertainty in the conjugate momentum density, requiring infinite energy in a zero-volume region. This violates BLID. Therefore, observables must be smeared over test functions f: ϕ(f) = Zd4x f(x)ϕ(x), f ∈C∞ c. Only smeared operators represent finite-energy, finite-information operations compatible with local capacity bounds. Remark 3.A “field” is not a physical object vibrating at a point; it is the local information buffer required to store accessible information while supporting LAIE globally. 6 Canonical (Anti)Commutation Relations Locality, statistics, and causal independence imply: [ϕ(f), ϕ(g)]∓=i∆(f, g), where ∆ is the causal Pauli–Jordan distribution. Proof. To satisfy LAIE, operations in spacelike regions must commute (or anticommute). The commutator is a c-number (to preserve linearity of independent additions). Relativistic covariance (Paper III) fixes the kernel to be the unique causal Green’s function. 7 Vacuum, Normal Ordering, and Operational Grounding Definition 1 (Operational vacuum).The vacuum |0⟩is the state minimizing the cost of creating distinguishable records under BLID. Remark 4.Vacuum energy is not an absolute quantity but a reference calibration. Normal ordering is the operational subtraction of the baseline information cost required to maintain the buffer. It is not an ad hoc trick, but the definition of the zero-information state. – 3 – 8 Conclusion Quantum fields are not fundamental entities postulated by hand. They are the unique mathematical structures capable of reconciling: 1. Unlimited global excitation number (LAIE), 2. Finite local operational resolution (BLID), 3. Locality and statistics. Fields are local information buffers. Fock space is the only valid global bookkeeping scheme. This completes the kinematical reconstruction. Paper VIII will derive gauge redundancy as operational equivalence. – 4 –