"Quantum mechanics is just thermodynamics in imaginary time." It...
Jonathan Gorard@getjonwithit
22 views
Aug 12, 2024
~3 min read
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"Quantum mechanics is just thermodynamics in imaginary time."
It sounds like a hand-wavy, quasi-philosophical statement. But with a brief dive into the relationship(s) between hyperbolic and parabolic PDEs, it becomes possible to formalize it mathematically. (1/16)
It sounds like a hand-wavy, quasi-philosophical statement. But with a brief dive into the relationship(s) between hyperbolic and parabolic PDEs, it becomes possible to formalize it mathematically. (1/16)
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(2nd-order) PDEs may be classified broadly as hyperbolic, elliptic or parabolic, based upon their relationship with time and causality. Hyperbolic PDEs have a notion of time/causality: a perturbation somewhere in the domain propagates out, wave-like, with a finite speed. (2/16)
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Elliptic PDEs have no notion of time/causality: a perturbation somewhere in the domain affects everywhere instantly, with an effectively infinite speed of information propagation. Parabolic PDEs are a strange intermediate: they have a notion of time, but not of causality. (3/16)
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A perturbation somewhere in the domain of a parabolic PDE still affects everywhere instantaneously (like an elliptic PDE), but the perturbation still "diffuses" out over time. We can make this characterization mathematically precise by reducing 2nd-order PDEs to coupled... (4/16)
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...systems of 1st-order PDEs. The characteristics (which determine causality/the flow of information) of the 2nd-order PDE are then given by the eigenvalues of the 2x2 matrix describing the coupled system of 1st-order PDEs. As with all solutions to a quadratic equation... (5/16)
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...this means either two distinct real eigenvalues, no real eigenvalues (i.e. two distinct complex eigenvalues), or one repeated/degenerate real eigenvalue. In the case of two distinct real eigenvalues, we have two characteristic lines, which join to form a spacetime cone. (6/16)
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This cone represents the past/future "light cone" of each point in the domain, thus representing a hyperbolic PDE with a finite speed of information propagation. In the case of no real eigenvalues (i.e. two distinct complex eigenvalues), we have no characteristic lines,... (7/16)
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...thus representing an elliptic PDE with an unbounded speed of information propagation. In the case of a single/degenerate real eigenvalue, we have a single characteristic line, which represents a vertical line lying across the spacetime domain, thus representing... (8/16)
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...a parabolic PDE, in which every point in the domain depends upon all of the points lying in its past (without any light cone/causality structure), but not on any of the points in its future. In effect, the single characteristic line neatly separates past from future. (9/16)
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By transforming one of the variables in a PDE to be imaginary (e.g. making a t -> it transformation), we can force the discriminant of this quadratic to change sign. In particular, we can force a previously parabolic PDE to become hyperbolic (or indeed elliptic). (10/16)
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If we apply this t -> it transformation to the heat equation - i.e. the prototypical example of a parabolic PDE, representing the diffusion of heat in thermodynamics - we indeed obtain a hyperbolic PDE, by forcing the previously degenerate eigenvalues to become distinct. (11/16)
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But not just any hyperbolic PDE: the Schrödinger equation, describing the evolution of the wave function in quantum mechanics. This t -> it transformation is known as a Wick rotation, and it thus allows one to transform quantum mechanical problems into thermodynamic ones. (12/16)
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(Or vice versa). This explains why the same basic kinds of gadgets (e.g. correlation functions, partition functions, etc.) appear both in quantum mechanics and in statistical mechanics, despite the two theories being apparently distinct, and also "explains"... (13/16)
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...at least mathematically, if not physically, why the exp(-iHt) term that appears in the Schrödinger equation looks so formally similar to the exp(-ßH) term that appears in the Gibbs measure: they're really the same term, just related by a t -> it transformation. (14/16)
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The duality is so deep that it persists into quantum field theory too, via the Osterwalder-Schrader theorem, which rigorously relates quantum and statistical field theories. There is still much that is not understood about this formal analogy, but underneath it all... (15/16)
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...is a basic feature of 2nd-order PDEs: that diffusion in real time is related to wave propagation in imaginary time, and vice versa. (16/16)
