Logical Quantum Field Theory
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We consider the statistical mechanical ensemble of bit string histories that are computed by a universal Turing machine. The role of the energy is played by the program size. We show that this ensemble has a first-order phase transition at a critical temperature, at which the partition function equals Chaitin's halting probability Ω. This phase transition is almost zeroth-order in the sense that the free energy is continuous near the critical temperature, but almost jumps: it converges more slowly to its finite critical value than any computable function. We define a non-universal Turing machine that approximates this behavior of the partition function in a computable way by a super-logarithmic singularity, and study its statistical mechanical properties. For universal Turing machines, we conjecture that the ensemble of bit string histories at the critical temperature can be formulated as a string theory.
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