# Meaning of "symbol" in algebraic field theory?

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• strangerep
In summary, the conversation discusses the use of the term "symbol" in the paper "Causal Lie products of free fields and the emergence of quantum field theory" by Bucholtz, Longo, and Rehren. The term refers to a set of elements indexed by elements of the Schwartz space and used to generate a Lie algebra. The conversation also touches on the assumption of ##\phi## as an operator-valued distribution and its relationship to quantum fields.
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?

##\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).

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Oh, thanks, I was over-thinking it.

"Simples".

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).

martinbn said:
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.

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strangerep said:
[...].

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.

## What is the meaning of a symbol in algebraic field theory?

In algebraic field theory, a symbol is a mathematical notation that represents a specific element or concept in the field. It is used to simplify and generalize mathematical expressions and equations.

## How are symbols used in algebraic field theory?

Symbols are used in algebraic field theory to represent variables, constants, and operations. They are also used to denote relationships between different elements in the field.

## What is the difference between a symbol and a variable in algebraic field theory?

In algebraic field theory, a symbol is a general notation that can represent any element or concept in the field, while a variable is a specific symbol that represents an unknown quantity. Variables can be assigned specific values, while symbols remain general.

## Can symbols have different meanings in different contexts in algebraic field theory?

Yes, symbols can have different meanings in different contexts in algebraic field theory. For example, the symbol "x" can represent a variable in one equation, but a constant in another equation.

## How do symbols contribute to the understanding of algebraic field theory?

Symbols play a crucial role in algebraic field theory by providing a concise and standardized way to represent mathematical concepts and relationships. They allow for complex equations and expressions to be written and manipulated in a more efficient and systematic manner.

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