Quantum Yang-Mills theory, Classical Statistical Field Theory (for Hamiltonians which are non-polynomial in the fields, e.g. General relativistic statistical mechanics) and Quantum Gravity all suffer from severe mathematical inconsistencies and produce unreliable predictions at best.
We define a class of statistical field theories where the probability distribution of the infinite-dimensional phase-space is mathematically defined by a wave-function in a Fock-space, allowing Hamiltonians which are non-polynomial in the fields. In this formalism, the wave-function is crucial already for classical statistical field theory due to mathematical reasons.
It was often claimed that the mathematical inconsistencies only affect Field Theory and spare Quantum Mechanics (which would only suffer a mild interpretation ambiguity) but here we show that instead the mathematical inconsistencies from Field Theory do have collateral consequences for the ensemble interpretation of Quantum Mechanics (where Quantum Mechanics merely describes a statistical ensemble of objects and not individual objects).
We also show that for a quantum system defined in a 2-dimensional real Hilbert space, the role of the (2-dimensional real) wave-function is identical to the role of the Euler's formula in engineering, while the collapse of the wave-function is identical to selecting the real part of a complex number; the collapse of the wave-function of any quantum system is a recursion of collapses of 2-dimensional real wave-functions.
We then define a subclass of statistical field theories in Minkowski space-time where the (classical) canonical coordinates when modified by a non-deterministic time evolution, verify the canonical commutation relations. By defining gauge symmetries through algebraic ideals, we propose definitions for Quantum Yang-Mills and Quantum Gravity. Finally, we test the consistency of our formalism with the quantization of the free Electromagnetic field.
*Chapters ending with an * symbol address foundational questions and can be skipped in a first reading.
Introduction to the book "Timepiece: Mathematical inconsistencies of Statistical Field Theory and its consequences for Quantum Mechanics and Machine Learning".
We show that the classical initial value problem x'=x^2 has a continuous solution for all time, when the initial x has any finite amount of uncertainties (which is unavoidable in engineering). This is in stark contrast with the singular solution when x has no uncertainties.
We show that any conditional probability measure in a standard measure space is parametrized by a unitary operator on a separable Hilbert space. We propose unitary inference, a generalization of Bayesian inference. We study implications for classical statistical mechanics.
We show that the wave-function parametrization allows to implement gauge symmetries as exact constraints in a standard probability space without attributing to the constrained space null-measure. The wave-function parametrization of classical dissipative dynamics is also studied.
Definition of a class of statistical field theories where the probability distribution of the infinite-dimensional phase-space is mathematically defined by a wave-function in a Fock-space, allowing Hamiltonians which are non-polynomial in the fields.
The wave-function parametrization is so important because it allows us to define mathematically a general probability measure for an infinite number of random variables. This renders Bayesian inference as a general and theoretically grounded framework for Machine Learning.
Using the simple fact described in the title, we prove the existence of a computational problem with implications to Machine Learning, Quantum Mechanics and Complexity Theory. We also prove P!=NP, but this claim still needs to be reviewed by experts in Complexity Theory.
In Bayesian inference there is always a prior probability distribution, and there is no prior which is better for all cases: we always have to make assumptions. This has profound implications for the concepts of consciousness, and extraterrestrial and artificial intelligences.
We show that a unitary quantum time-evolution defines the random sampling of an irreversible deterministic time-evolution. The coexistence of time-symmetric laws of physics and an arrow of time, is due to unitarity and not entropy. We also study the baryon asymmetry.
We show that the alignment problem in deep learning can be separated in two: A) estimate systematic uncertainties as in Engineering; B) new due to Big Data, how to randomly sample computable models from a set of Bayesian models, almost all uncomputable. Deep neural nets solve B).
