Monday, March 30, 2020

The quantum arrow of causality

A new paper argues that Quantum causality determines the arrow of time:
Eddington introduced the concept of the arrow of time - the one way flow of time as events develop and our perceptions evolve. He pointed out that the origin of such an arrow appears to be a mystery in that the underlying laws of physics (at least at the time of Eddington) are time symmetric and would work equally well if run in the reverse time direction. The laws of classical physics follow from the minimization of the action and are indeed time symmetric. This view has been beautifully captured by Carlo Rovelli [1], who writes: "The difference between past and future, between cause and effect, between memory and hope, between regret and the elementary laws that describe the mechanisms of the world, there is no such difference." But what picks out only those solutions running forward in time?

By now, there is a large literature on the arrow of time [1-10]. Essentially all of the literature accepts the proposition that the fundamental laws of physics do not distinguish between past and future and could equally well be run backwards. There is also a recognition that the second law of thermodynamics does distinguish between these directions as it states that entropy cannot decrease in what we refer to as the future. This leads to the idea of a thermodynamic arrow of time. Many view this thermodynamic arrow as the origin of the passage of time, or at least of our consciousness of that passage.

Our point in this paper is that the basic premise of such reasoning is not valid in quantum theory. Quantum physics in its usual form has a definite arrow of causality - the time direction that causal quantum processes occur. ...

The basic statement saying that the fundamental laws of physics do not differentiate an arrow of time is not correct. At the microscopic level, reactions run in one direction but not the other.
I think this paper is correct in that (1) a common view today is that the thermodynamic arrow of time is the only one; and (2) this view is mistaken.


  1. 1. Proving the existence of the arrow of time is problematic only if you take just one essential part of calculus, viz., (formulating) differential equations, and illogically *elevate* it over the other equally important essential, viz., specifying auxiliary data and solving the equation using techniques of integration.

    Consider the simplest case. Suppose the governing differential equation is dx/dt = k, where k is a positive constant. That's Newton's first law for you.

    They say that you can't see the arrow of time in this equation because, according to them, it is clear that the law is equally valid in both the forward and backward directions of time. (You can march the solution equally well in either time-direction, they say.)

    But saying so is glossing over the fact that if the initial condition at t = t_1 is x_1 (position) = 0, then the final position x_2 is *always* going to be a *positive* value, *if* the time parameter is taken to increase in the solution procedure. OTOH, x_2 is *always* going to be negative if t decreases in the solution procedure (which, physically means, we project the position from where the particle came to the x=0 point).

    The respective positive-ness and negative-ness for the x_2 values are completely determined by the law *and* the solution procedure involved (IC/BCs and integration techniques).

    Seen another way, if you recognize the backward march in the ***mathematical solution*** procedure as nothing but a conceptual-level/abstract projection (a flip) of the forward march, then it is clear that ***physically***, the universe always evolves only in the forward direction.

    People gloss over such facts, and then try to hunt for *differential* equations that show asymmetry in solutions regardless of their physical roots or applications. But this is a basically wrong way to approach the whole thing. ... Who cares for the arrow of time in maths alone, as torn from its roots in, and applications to, physics?

    2. Finally, even if you had to commit the above-mentioned error, still, an alternative more basic to both QM and thermodynamical (entropy-based) definitions are possible. Just follow the historical approach---tracing it backwards in time. Go to the diffusion equation. Fourier formulated and solved it in between 1804--1807. (Sadi Carnot's research was around early 1820's.)

    In diffusion, the backward march is *provably* ill-posed in the Hadamard sense---because of the combination of the Laplacian operator for space and the first-order differential for time. That's nothing but the arrow of time for you!

    To see the most crucial part of the proof, use finite differences, and observe that at every time-step, you add the contributions coming in from all the neighbours. Addition destroys the information of which neighbour contributed how much. Voila! (If you cache the information, the memory requirement goes on increasing. Effectively, you are introducing ways of manipulating information other than what the paradigm itself defines.)

    The thermodynamic and QM definitions are nothing but this same simple fact, but formulated using two progressively higher-level ontological viewpoints.

    3. To conclude, epistemologically, the crucial flaw to recognize is the fact that the very formulation of this "problem" (i.e., looking for an arrow of time in the differential equations alone, is deeply problematic, because it drops out of the context one half of calculus.

    Hierarchically, *both* differentiation and integration come at the same time, and they both are essential for making completely meaningful statements.

    But then, physicists (and philosophers of physics, and (too) many IT industry people) are what they are. They love empty intellectualizations. So, best policy, IMO, is to state your case and leave it at that. People looking forward to do empty intellectualization are like bottomless vessels. Nothing holds.


  2. Hearing people rhapsodize over there being no fixed direction of time while ignoring the fact their own precious math is utterly dependent upon fixed orders of logical operation is irony at its finest.

    Simple evaluation tool: If your precious theory depends upon a logical process outside of time to allow it to function (i.e. meta-time and other assorted ilk), it's wrong.