Recently I came across several elucidations about symmetry and conservation laws, within which numerous errors occur. So I write this post to make some clarifications.

Before proceeding this, we need to retrospect the famous Noether’s Theorem:

Every continuous symmetry of theory corresponds to a conservation law.

We hope you can recite every word carefully before reading the following content, since this is a mathematical theorem and no mathematical statements contain redundant words.

Symmetry of Matter and Conservation Law

There are no direct links between symmetry of matter field and conservation laws

The relationship between matter field and conservation law is one of the leading fallacies in Physics. The correspondence of symmetry and conservation laws has been introduced widely in physical courses. However, it seems that not all of them are able to illustrate this precisely. As a matter of fact, according to the Noether’s Theorem we invoked above, only the symmetry of theory can result in a conservation law, which means at least Noether’s Theorem can not help to draw a conclusion on anything related to the symmetry of matter. Mixing symmetry of theory and that of matter is a traditional mistake especially in condensed matter Physics. This has led to a number of advanced misleading statements, for example the famous more is different.

CPT Symmetry and Conservation Law

CPT symmetry can not lead to conservation laws according to Noether’s Theorem.

CPT symmetry is a concept in Quantum Field Theory, which is the abbreviation of charge, parity and time symmetry. In our previous featured article Structure of Physics, we predicate that no conservation laws can be falsified experimentally. Later on, I received a “counterexample” which is the breaking of parity conservation in electroweak interaction. Truly, in electroweak interaction, the parity symmetry breaks and the so-called parity conservation “corresponding to” it disappears. Have you noticed the quotes I used here? If you remember Noether’s Theorem above, you will immediately find out that it requires the symmetry to be continuous. Nevertheless, CPT symmetry corresponds to a group like which is on the contrary discrete symmetry. Also you can see in our document, the symmetry that theorists really care is categorized into Galilean, Lorentz, general and internal symmetry, within which the very CPT symmetry is not. Thus, the failure of parity conservation brings no threats to our methodology.

Finally, we again recall Feynman’s words

I know very early about the difference between knowing the name of something and knowing something.