Time Symmetry

The fundamental equations of physics at least those that derive from symmetries in nature all exhibit time symmetry because they are second-order differential equations. Newton’s second law and Maxwell’s equations are immediate examples. However, one can consider time running forward or backward. Even general-relativity equations formulated in tensor mathematics exhibit time symmetry. Assuming that all these equations are correct, must nature at its most fundamental level obey time symmetry? (Note: Entropy relations are not derived from a fundamental symmetry and therefore are excluded.)

Answer

No, nature does not need to obey time symmetry at the most fundamental level. Equations sometimes have more symmetry than the actual underlying physics behavior. For example, even though the tensor equations of general relativity are time-symmetric, they can be derived from a more fundamental type of mathematical entity called a twistor. Twistor equations of general relativity are not time-symmetric. In applications to a black hole, for example, the tensor equations predict time symmetry, but the twistor ones do not. As a consequence, the formation of a black hole and the time reversed version cannot both represent real physical behavior.

One might think that the quantum theory described by the Schrodinger equation is time-asymmetric, the equation being first order in time. As Roger Penrose points out in the reference below, quantum theory and its equations are indeed time-asymmetric. The wave function can be used to calculate the probability of a future state on the basis of a known past state, but not the other way that is, one cannot calculate the probability of a past state on the basis of a future state. You cannot retrodict the past!