Meaning of "symbol" in algebraic field theory?

[strangerep]

TL;DR Summary

Physicist-friendly explanation of "symbol $\phi(f)$", please?

I'm probably inadequately equipped to understand this paper by Bucholtz, Longo and Rehren on "Causal Lie products of free fields and the emergence of quantum field theory", but I decided to give it a try. Alas, I got stuck in the 1st para of sect 2 where it says:

We consider a Lie algebra $\Phi$ that is generated by the symbols $\phi(f)$ which are real linear with regard to $f \in \mathcal{s}(\mathbb{R}^d)$. [...]

Although I've seen the term "symbol $\phi(f)$" before, I've never succeeded in properly understanding what it means. Could someone please explain the meaning of this use of "symbol" in a physicist-friendly way?

 

[Demystifier]

$\phi(f)=\int d^4x\, \phi(x)f(x)$
It's similar to a Fourier transform, except that $f(x)$ is not a plane wave but a function that better behaves under integrals (e.g. a function from a Schwartz space or a function defined on a compact support).

 

[strangerep]

Oh, thanks, I was over-thinking it.

"Simples".

 

[martinbn]

I might be misreading it but isn't it meant in the usual sense of the word? You just take the set of elements indexed by the elements of the Schwartz space, and denoted by $\phi(f)$, then consider the Lie algebra generated by them (with some additional requirements, that they are linear and satisfy the Klein-Gordan equation).

 

[strangerep]

I might be misreading it but isn't it meant in the usual sense of the word?

It depends what you mean by "usual". In standard English, it means:

noun:
1. something used for or regarded as representing something else; a material object representing something, often something immaterial; emblem, token, or sign.

2. a letter, figure, or other character or mark or a combination of letters or the like used to designate something: the algebraic symbol x; the chemical symbol Au.

3. (especially in semiotics) a word, phrase, image, or the like having a complex of associated meanings and perceived as having inherent value separable from that which is symbolized, as being part of that which is symbolized, and as performing its normal function of standing for or representing that which is symbolized: usually conceived as deriving its meaning chiefly from the structure in which it appears, and generally distinguished from a sign.

... none of which are helpful to understand the mathematical meaning.

Anyway,.. no worries... I get it now.

Still,... it's seem strange (to me anyway) that they use the phrase "emergence of QFT" in their title, but tacitly assume (ISTM) right from the start that $\phi$ is an operator-valued distribution.

 

[dextercioby]

[...].

Still,... it's seem strange (to me anyway) that they use the phrase "emergence of QFT" in their title, but tacitly assume (ISTM) right from the start that $\phi$ is an operator-value distribution.

No, not a Fock-state-operator-valued distribution. That is already a quantum field. The $\phi(f)$ should only be a distribution on (typically) Schwartz space.