Wightman axiom 2: what kind of representation?

The value of the classical field at a point in spacetime $phi(x)$ may transform under any finite-dimensional representation, not necessarily unitary or orthogonal etc.

But the quantum field as an operator-valued distribution transforms under an infinite-dimensional unitary representation that acts on the Hilbert space of the QFT.

Wightman axioms relate the two representations, postulating that

$$ U(Lambda) phi(f) U(Lambda)^{dagger} = P(Lambda) phi (S(Lambda^{-1}) f). $$

Here $U(Lambda)$ is the infinite-dimensional unitary representation, $P(Lambda)$ is the finite-dimensional representation that acts on the classical field's value at a point, and $S(Lambda)$ is the natural representation that acts on test functions over spacetime.

Well the answer here is that the author is talking about Irreducible representations (Irreps). But @Plop you have asked me a great question in the comments, "why physicists like so much irreducible representations?". So I done my best to answer it.

TLDR: Physicists primarily deal with Lie Groups (Poincare group is a Lie Group) and there is a theorem which states that "If the Lie group representation isn't already irreducible than it can be "completely reduced" into a collection of irreducible representations." So by the nature of our math tools we MUST be using Irreps

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I studied the Burau representation in undergrad (https://en.wikipedia.org/wiki/Burau_representation) and then quantuum chromodynamics in grad school, so I'll try to connect express how I have connected those two experiences.

1. What is representation theory? A represenation of a group $G$ is a homomorphic mapping of the group $G$ onto a non-singular group of $d times d$ matrjces $Gamma(T)$, where matrix multiplication is the group's multiplicative operation. (The group of matrices $Gamma(T)$ forms a $d$-dimensional representation $Gamma$ of group $G$)

Example I think that the unreduced Burau representation of the Braid Group $B_n$ (https://www.youtube.com/watch?v=uMMxD0Ak4lg) gives a great visual interpretation! This representation maps the act of crossing two strands of hair (left over right) $sigma_i$ onto the matrix, $$ begin{align} (0) && Gamma(sigma_i) = left[ begin{array}{c|cc|c} I_{i-1} & 0 & 0 & 0 hline 0 & 1-t & t & 0 0 & 1 & 0 & 0 hline 0 & 0 & 0 & I_{n-i-1} end{array} right] end{align} $$ By definition the representation connects to uncrossing two strands of hair (right over left) $sigma_i^{-1}$ with matrix $Gamma(sigma_i)^{-1}=Gamma(sigma^{-1})$.

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2. What are (ir)reducible representations? Reducible representations have the form $$ begin{align} label{eq_reducible} (1) && Gamma(T) = begin{bmatrix} Gamma_{11}(T) & Gamma_{12}(T) 0 & Gamma_{22}(T) end{bmatrix} end{align} $$. Such that $$ Gamma(T_1) Gamma(T_2) = left[ begin{array}{c|c} Gamma_{11}(T_1) Gamma_{11}(T_2) & Gamma_{11}(T_1) Gamma_{12}(T_2) + Gamma_{12}(T_1) Gamma_{22}(T_2) hline 0 & Gamma_{22}(T_1) Gamma_{22}(T_2) end{array} right] $$ We have arrived at the homomorphic property which defined a representation. $$Gamma_{11}(T_1 T_2) = Gamma_{11}(T_1) Gamma_{11}(T_2) $$ $$Gamma_{22}(T_1 T_2) = Gamma_{22}(T_1) Gamma_{22}(T_2) $$ Therefore the representation $Gamma$ contains representations $Gamma_{11}$ and $Gamma_{22}$ which are of a smaller dimension than $Gamma$. Since these smaller representations exist within $Gamma$, gamma is reducible. An irreducible group contains no such smaller dimensional representations.

Example Fortunately for us, the Reducible Burau representation Eq.0 has this structure! Consider $Gamma(sigma_1)$ and $Gamma(sigma_3)$ in $B_4$ $$ Gamma(sigma_1) Gamma(sigma_3) = left[ begin{array}{cc|cc} 1-t & t & 0 & 0 1 & 0 & 0 & 0 hline 0 & 0 & 1 & 0 0 & 0 & 0 & 1 end{array} right] left[ begin{array}{cc|cc} 1 & 0 & 0 & 0 0 & 1 & 0 & 0 hline 0 & 0 & 1-t & t 0 & 0 & 1 & 0 end{array} right] = left[ begin{array}{cc|cc} 1-t & t & 0 & 0 1 & 0 & 0 & 0 hline 0 & 0 & 1-t & t 0 & 0 & 1 & 0 end{array} right] $$ This is great and it indicates that the Unreduced Burau representation can be reduced! (https://en.wikipedia.org/wiki/Burau_representation#Explicit_matrices)

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3. How does the process of Reduction Happen?
   Suppose that $Gamma_{11}(T)$ is itself reducible and be expressed in the same for as Eq.1. Fortunately you can always similarity transform $Gamma(T)$ such that $$S^{-1} Gamma S = Gamma' = left[ begin{array}{cccc} Gamma'_{11}(T) & cdots & cdots & Gamma'_{1n}(T) 0 & ddots & cdots & Gamma'_{2n}(T_2) vdots & 0 & ddots & vdots 0 & cdots & 0 & Gamma'_{nn}(T) end{array} right]$$ If this upper triangular representation can be "completely reduced" to a block diagonal using another similarity transform, then $Gamma(T)$ is a completely reducible representation. There is a theorem which states that "if a Lie group is reducible it is completely reducible." - "Group Theory in Physics" Ch4-S4 J.F. Cornwell (1984).

$$text{CITATION: https://www.sciencedirect.com/topics/mathematics/reducible-representation}$$