long time ago, the post on parity is quit a mess and confusing. now, i have a better understanding and try to do it once again.

here, i will clarify 2 things:

1. what is parity.
2. what will happen if parity is hold or violated.

the answer of 1st question is

Parity is a transform that reverse the space dimension.

it sound simple. the tricky part is, there are 2 kinds of transform: active or passive. in short, active transform is change of vector. while pass is change of coordinate. To avoid confusion and make me and reader thinking to be the same, i use active transform in here.

in 1-D space, the parity is just reverse of the direction. in 2-D space, parity is equal a rotation of 180 degree on the plane, thus, it is not really a parity but as a special case for rotation. in 3-D space, parity is just like changing our left-hand to right hand and vice versa. many people will say that, it is same as MIRROR transform, which is a plan inverse. But it is not true. the correct description is a mirror transform + 180 rotation around the normal of the mirror plan. Thus, if we only concern about the shape instead of orientation, a mirror transform is ok. Nevertheless, when we talking current loop, the direction is important that, the magnet is a mirrored inside the mirror. and we knew that all magnetic field is either by current loop or spin.  in 4-D, well, i don’t know.

To answer the 2nd question, we have to know what is parity conservation. the parity conservation is that, thing keep the same under parity transform. i.e.

$P \Psi(x) = \Psi(-x) = \pm \Psi(x)$

the plus sign means EVEN parity, and minus sign means ODD parity. people knowing some maths will know that, it is corresponding to even function and odd function.

Lets look at the parity of electromagnetic phenomena. lets there is a current loop, rotating at left hand. the B field will be pointing down. under MIRROR (not parity) by building an identical current loop which rotates in right hand, then the magnetic field pointing upward. Wait! is it mean magnetic field violated parity? NO, because it is the pseudovector nature of magnetic field, which, keep the direction perpendicular to the plan of mirror and change sign on parallel to the mirror plan. and as i said before, a parity is mirror + 180 rotation. thus, the parity is conserved. (polar vector has opposites properties that, under mirror, the component parallel to mirror plan is kept but the component perpendicular reversed. well this is what mirror transform was defined)

Now, we found that, all pseudovector are even parity and polar vector are odd parity. (although i don’t have a prove, i believe this is strongly related to spin1/2 and spin 1, and every pseudovector is formed by polar vector. )

Thus, a parity conservation mean that, all physics are same under parity. The other thing to say so, is, the mirrored world is same after we rotated it back.

$P=MR=RM \Rightarrow RP=PR=M$

in order to understood what is parity conserved or violated, we study elastics scattering. if we only have polar vector, a parity violation means the space has preference direction, i.e. the symmetry broken by a pseudovector field. but that contradict our assumption of “only polar vector”. thus, it is impossible to have parity violation for polar vectors. (i am not sure it is a prove or not)

A famous example of parity violation in nature is the experiment on Cobalt-60 nuclei, which will undergoes beta decay.

*** at the beginning, i was very confused. one major confusion is that, the active transform or passive transform and the mirror world. i found that, inside the mirror, we should NOT use “left-hand rule” instead of  “right-hand rule”.  if we do a mirror transform, and use “left-hand rule” , the transform will be cancelled. because, changing the “handiness” is same as passive transform.

another difficulty is that, a mirror transform is not parity transform, but most popular science mixed them up. a Major difference between mirror and parity is: there are 3 difference reflection plans, but parity is a “point inversion”. when you select difference plan of reflection, your current loop will keep the same or change direction, that make it sometimes keep orientation, but sometimes not.

that difficulty raised because lack of understanding the proper transform of pesudo vector under mirror transform. we the help of the picture, we can now easily imagine parity as mirror.