# Is it legal to add two physical quantities that behave differently under time-reversal, parity etc to define a new that has no definite behaviour?

In physics, can we legally add two physical quantities that behave differently under time-reversal or parity? For example, let $$\vec{a}$$ and $$\vec{b}$$ are two observables. Let $$\vec{a}$$ flips sign under time-reversal (or parity) but $$\vec{b}$$ does not. For example, $$\vec{a}$$ is velocity and $$\vec{b}$$ is the angular momentum of a particle. Can we legally add them (of course, by suitable making their dimensions the same) to define a new observable $$\vec{c}=\vec{a}+\alpha\vec{b}$$ (where $$\alpha$$ is some appropriate dimensional constant) such that it has no definite behaviour under time-reversal (or parity)? If this is illegal or meaningless, I would like to see/know why. Thanks.

• an interesting question. Well, certain behaviour under parity and time-reversal is not neccessary to be imposed from the philosphical point of view. However, i a not aware of any sensible quantity, composed as a sum of vector and pseudovector Jul 30, 2020 at 6:44
• @ChiralAnomaly I think the ocalMathmatician's answer and the comments are good enough but I had another thing in mind: the 3-vector part of the Pauli-Lubanski vector $W_\mu$ is $\vec{W}=E\vec{J}-\vec{P}\times \vec{K}$ and it left me to wonder whether this quantity $\vec{W}$ must have to have definite parity/time-reversal behaviour. en.wikipedia.org/wiki/… Aug 1, 2020 at 17:36

What I'm talking about is the Fermi V-A theory of weak interactions. From the experimental results of the time, Fermi built up the first theory of weak decays, which we now know as to be mediated by the weak interaction, through the following hamiltonian $$\mathcal{H}_F = -\frac{G_F}{\sqrt{2}}\bar{\nu}_\mu\gamma^\sigma(1-\gamma_5)\mu\,\bar{e}^-\gamma_\sigma(1-\gamma_5)\nu_e\tag{1}$$ Although Fermi firstly hypothesized the hamiltonian for the decay of the neutron, I gave you the hamiltonian for the decay of the muon since it has a simpler form and it's easier to see that the hamiltonian $$(1)$$ is build up from the difference of a vector quantity and a pseudovector quantity. Given that $$\nu_\mu$$, $$\mu$$, $$e^-$$, $$\nu_e$$ are all spinors of the related particle, which I'll now call simply by a general $$\psi$$, we know that $$\bar\psi\gamma^\mu\psi\qquad\text{Vector}\\\bar{\psi}\gamma^\mu\gamma_5\psi\qquad\text{Pseudovector}$$ and it's easy to see that the Fermi hamiltonian contains two such quantities $$\bar{\psi}\gamma^\mu\psi-\bar{\psi}\gamma^\mu\gamma_5\psi$$ which is a difference of a vector and a pseudovector.