Quantum field theory unifies special relativity and quantum mechanics yet relies on externally imposed prescriptions—quantization, renormalization, and probabilistic interpretation—that lack a common algebraic foundation. This work derives these features from a single structural principle: the finite-resolution realization of the additive group [Formula: see text] under four internal consistency constraints. These constraints—finite operational resolution (FR1), multiplicative linearity (FR2), phase coherence (FR3), and statistical typicality (FR4)—generate, without additional postulates, the Hilbert-space formalism of quantum theory, the Born probability rule, and the Kolmogorov structure of statistical mechanics as limiting projections of one additive algebra. When extended to spacetime translations and their automorphisms, the same framework reproduces the kinematic and dynamical structure of quantum field theory while resolving ultraviolet divergences through intrinsic finiteness rather than external renormalization. Quantization and unitarity appear as consequences of additive symmetry, while the gravitational field emerges as the algebraic kernel that transports boundary information into bulk observables without requiring a separate quantization of the metric. The result is a unified, finite, and strictly algebraic foundation for physical law.
There is a well-known analogy between statistical and quantum mechanics. In statistical mechanics, Boltzmann realized that the probability for a system in thermal equilibrium to occupy a given state is proportional to
exp(−E/kT), where
E is the energy of that state. In quantum mechanics, Feynman realized that the amplitude for a system to undergo a given history is proportional to
exp(−S/iℏ), where
S is the action of that history. In statistical mechanics, we can recover Boltzmann's formula by maximizing entropy subject to a constraint on the expected energy. This raises the question: what is the quantum mechanical analogue of entropy? We give a formula for this quantity, which we call ``quantropy''. We recover Feynman's formula from assuming that histories have complex amplitudes, that these amplitudes sum to one and that the amplitudes give a stationary point of quantropy subject to a constraint on the expected action. Alternatively, we can assume the amplitudes sum to one and that they give a stationary point of a quantity that we call ``free action'', which is analogous to free energy in statistical mechanics. We compute the quantropy, expected action and free action for a free particle and draw some conclusions from the results.