Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ward - Takahashi identities and Noether's theorem in quantum field
  theory   A gap in the mathematical logic in derivations in quantum field theory arises
as consequence of variation before quantization. To close this gap the present
paper introduces a mathematically rigorous variational calculus for operator
fields. Using quantization before variation it is demonstrated that the
so-called naive results are correct; in particular both Noether's theorem and
the Ward-Takahashi identities retain full validity in quantum field theory.

24
The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex