Tuesday, January 19, 2016

lagrangian formalism - Does a constant factor matter in the definition of the Noether current?


This is a very basic Lagrangian Field Theory question, it is about a definition convention. It takes much more time to typeset it than answering, but here it is:



Consider a field Lagrangian with only a kinetic term,


L=12μϕμϕ


Consider the very simple transformation ϕϕ+α (α constant), and so I understand here that α plays the role of δϕ. I determine the Noether current as L[μϕ]δϕ


and the result is αμϕ


But in Peskin & Schroeder (just above eq 2.14), the result they give is:


μϕ


And it doesn't seem to be an erratum. I don't care that "localized" Lagrangian very much (hey, wait before closing, please), but a very general question arises:


Is α dropped simply because μϕ is too a conserved quantity (and so under "conserved current" one understands the general concept, momentum, energy or whatever, regardless of its value), or am I missing some other very basic detail that is assumed to be known by the reader?




Later edit: I have eventually understood this question and more, by reading the beginning of chapter 22 of Srednicki. I am finding that book (well, the free preprint for the moment) crystal clear, it seems excellent.




Answer



I) Let us for simplicity address OP's question in the context of point mechanics where qi are generalized position coordinates on some manifold M [instead of considering field theory with fields ϕα(x)]. OP's question is rooted in the difference between




  1. on one hand, an infinitesimal variation qi ˜qi = qi+δqi

    of the generalized position coordinates, or equivalently, δq := ˜qiqi;




  2. and on the other hand, that of a generator/Lie algebra element/vector field Y = Yiqi,Yi = Yi(q),

    which is not infinitesimal (although Y is sometimes confusingly referred to as an 'infinitesimal generator' in the literature).





Both concepts δ and Y are linear derivations that satisfy Leibniz rule, and the interrelation between the two is given by


δqi = ϵYi,


where ϵ in eq. (4) is an infinitesimal parameter. The mathematical concept of a vector field Y is tied in a bijective manner to the concept of a flow1


σ: ]c,c[ × M  M,]c,c[  R,


where


ddϵσi(ϵ,q) = Yi(σ(ϵ,q)),σi(ϵ=0,q) = qi.

A flow σ satisfies σi(ϵ,σ(ϵ,q)) = σi(ϵ+ϵ,q).
Note that in eq. (7), it is understood that ϵ and ϵ are real numbers in the interval ]c2,c2[R, and not infinitesimal.


II) The (bare) Noether charge


Q = piYi


is (in this case) momentum


pi := L˙qi



times generator Yi. In particular, the definition (5) of the Noether charge Q does not depend on the ϵ parameter.


--


1 We ignore the possibility that the domain ]c,c[ could depend on the initial position qM.


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