I am struggling with equation 1.5 in Tong's QFT course. I try to understand/explain it in strict calculus, i.e. without physics shortcuts like "small variations". I guess in the full blown explanation, [itex]\delta S[/itex] is a total derivative.(adsbygoogle = window.adsbygoogle || []).push({});

To be specific, with total derivative I mean the linear map that best approximates a given function [itex]f[/itex] at a given point. For [itex]f:ℝ\toℝ[/itex] we have [itex]D(f,x_0):ℝ\toℝ[/itex], i.e. [itex]D(f,x_0)(h) \in ℝ[/itex]. Often it is also denoted as just [itex]\delta f[/itex].

In terms of total derivative, I wonder if the following simplification of Tong's equation 1.5 still catches what is going in in mathematical terms (no longer physical). Let [itex]S_{a,b}(f) = \int_a^b f(x)dx[/itex] a functional that maps functions [itex]f[/itex] to the real line. Then [itex]D(S_{a,b},f) = \delta S_{a,b}[/itex] should be well defined given any necessary smoothness conditions. In particular [itex]D(S_{a,b},f)[/itex] maps functions [itex]h[/itex] of the same type of [itex]f[/itex] to real numbers. Because the integral is linear, so my hunch, its best linear approximation should be itself. InTong's course, equation 1.5, first line, I find what I understand to be

[tex]\delta \int_a^b f(x) dx = \int_a^b \delta f dx[/tex]

Can anyone explain how the algebraic types on the left and on the right would match up? My interpretation is, that on the left I have a the total derivative of a functional, which itself should be a functional, written explicitly as [itex]D(S_{a,b},f)[/itex]. On the right I have the integral over, hmm, the total derivative of [itex]f[/itex], where I don't see how this could be a functional?

Any hints appreciated.

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# Quantum Field Theory lecture by Tong (Cambridg)

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