Why isn't the world symmetric under parity transformations?

Question

In quantum physics and newtonian mechanics, switching "left" and "right" in the math is super easy: you just state that left is, for instance, the positive $x$ axis and right is the negative $x$ axis, and then you switch up the signs of $x$ in the differential equation you want to solve and the initial conditions or boundary conditions you're applying to get a perfectly symmetric prediction. But if the laws of physics don't differentiate between right and left, and the world follows the laws of physics, why isn't the world symmetric? Most people write with their right hands, the right and left brain are different, and the organs of the human body are distributed unevenly between our right and left insides. But if the microscopic processes that take place in the fancy biological processes that let humans grow into full organisms from just a couple of cells are mathematical symmetric under parity, why are we asymmetric?

One of the suggested similar questions was this: Why isn't our universe symmetric?. I guess I understand what John Rennie's answer states: he implies that a tiny fluctuation in the early universe gave rise to asymmetry. Does that imply that there is a domino effect of sorts that leads to large-scale macroscopic asymmetry from small asymmetric fluctuations? Furthermore, doesn't the occurrence of that fluctuation as described by the theory of inflation (which I first heard of while reading that answer) imply breaking of parity symmetry?

Answer 1

The fact that a particular system has a symmetry does not imply that every state of the system has the same symmetry. To understand this, it's worth taking a moment to consider what it means for a system to obey a symmetry in the first place.

Consider a free particle on a line. This system has (among other things) left-right reflection symmetry. In quantum mechanics, this is reflected by the fact that the Hamiltonian commutes with the parity operator; in classical mechanics, it is reflected by the fact that the Hamilton equations are invariant under the exchange $x\mapsto x'=-x$.

But this does not mean that every state of the system has parity invariance. After all, for a classical particle there is only one parity-invariant state, namely sitting still at the coordinate origin; for a quantum particle, the space of parity-invariant states is larger, but is obviously not exhaustive. A wave packet with non-zero group velocity in either direction is an example of a state which does not feature parity invariance.

When we say that a system possesses parity symmetry, what we mean is that if we consider some $\gamma(t)$ which is a solution to the equations of motion, then the parity transformed trajectory is also a solution to the equations of motion. In other words, it's not that the system must be in a state which exhibits parity invariance; it's that the parity-transformed "partner" of any state must also be a valid solution to the equations of motion.

In broader terms, the universe possessing a certain symmetry does not imply that every state of the universe possesses the same symmetry, but rather that the state of the universe obtained via that symmetry transformation is also permitted and evolves in the same way.

But if the microscopic processes that take place in the fancy biological processes that let humans grow into full organisms from just a couple of cells are mathematical symmetric under parity, why are we asymmetric?

There are complicated biological and evolutionary reasons for human bodies being as they are. However, note that an organism which is exactly like a human except for being a mirror image (i.e. heart on the right, typically left-handed, etc) is not forbidden by the laws of physics, so there's no contradiction.

Answer 2

As it has already stressed in Murray's answer, the symmetry of the physical laws has not to be the same as the symmetry of the states.

However, as an additional piece of information, I would note that the world is not symmetric under parity transformation. This is the result of a famous experiment conducted in 1956 and showing that conservation of parity is violated in weak interactions. Therefore, the laws of Physics do differentiate between right and left on the scale of energies of those experiments. Interestingly, the parity symmetry is restored at the coarse-grained level of description, where the relevant degrees of freedom are (stable) nuclei and electrons.