7 Axiom 5 (Lorentz Covariance)

Next let us consider Axiom 5 (Lorentz Covariance)

Axiom 5 (Lorentz Covariance) The inhomogeneous Lorentz group is represented by automorphisms \(A \in \mathfrak {U} \rightarrow A^L \in \mathfrak {U}\) such that

\begin{align} \mathfrak {U}(\mathbf{B})^L = \mathfrak {U}(\mathbf{LB}) \end{align}

where \(L\mathbf{B}\) is the image of the region \(\mathbf{B}\) under the Lorentz transformation \(L\).

This axiom is the most straightforward axiom of all that we have encountered.

7.1 Inhomogeneous Lorentz Group (Connected to the Identity)

This original formulation of Axiom 5 (Lorentz Covariance) (Haag and Kastler) uses the term “inhomogeneous Lorentz group” without specifying if this refers to all connected components of the inhomogeneous Lorentz group or simply the component of the inhomogeneous Lorentz group that is connected to the identity. This is likely a simple oversight.

The components of the inhomogeneous Lorentz group not connected to the identity are obtained from the component connected to the identity by reflections in time and/or space (See Section I.2.1 of Haag), and it is an open question as to whether such reflections correspond to “physical” symmetries (Again see Section I.2.1 of Haag).

That being said, it is a reasonable assumption to make that the phrase “inhomogeneous Lorentz group” used in Axiom 5 (Lorentz Covariance) should more properly be read as “the component of the inhomogeneous Lorentz group connected to the identity”. We will adopt this posture.

7.2 Action of the Inhomogeneous Lorentz Group’s Identity Component

Consider an element \(L\) of the inhomogeneous Lorentz group’s identity component. For a given \(\mathfrak {U}(\mathbf{B})\) the action of \(L\) on \(\mathfrak {U}(\mathbf{B})\) should “naturally” take the form

\begin{align} \mathfrak {U}(\mathbf{B}) \rightarrow \mathfrak {U}(\mathbf{LB}), \end{align}

where \(\mathbf{LB}\) is the set that results from \(L\) acting on \(\mathbf{B}\).

By definition the set-theoretic union of all \(\mathfrak {U}(\mathbf{B})\) is a dense subset of the quasilocal algebra \(\mathfrak {U}\). Hence, this “natural” action of the inhomogeneous Lorentz group’s identity component on elements of the form \(\mathfrak {U}(\mathbf{B})\) defines its action on a dense subset of the quasilocal algebra \(\mathfrak {U}\).

This then leads to the question: How does one extend the action of the inhomogeneous Lorentz group’s identity component to all of the quasilocal algebra \(\mathfrak {U}\)?

It turns out, one can extend the action of the inhomogeneous Lorentz group’s identity component to all of the quasilocal algebra \(\mathfrak {U}\) by using the following theorem (Theorem A.36 Hall)

Theorem 4 Bounded Linear Transformation Theorem

Let \(V_1\) be a normed space and \(V_2\) a Banach space. Suppose \(W\) is a dense subspace of \(V_1\) and \(T: W \rightarrow V_2\) is a bounded linear map. Then there exists a unique bounded linear map \(\widetilde{T}: V_1 \rightarrow V_2\) such that \(\widetilde{T}|_W = T\). Furthermore, the norm of \(\widetilde{T}\) equals the norm of \(T\).

However, before applying this theorem to the case at hand we have to recount some properties of the elements we are working with.

First, as the quasilocal algebra \(\mathfrak {U}\) is a C*-algebra, it is a normed space and can play the role of \(V_1\) in this theorem.

Second, as the set-theoretic union of all \(\mathfrak {U}(\mathbf{B})\) is a dense subset of the quasilocal algebra \(\mathfrak {U}\), it can play the role of \(W\) in this theorem.

Next, by construction the quasilocal algebra is the completion of the set-theoretic union of all \(\mathfrak {U}(\mathbf{B})\), thus it is a Banach space. Hence, it can also play the role of \(V_2\) in the theorem.

Finally, the action of the inhomogeneous Lorentz group’s identity component on the set-theoretic union of all \(\mathfrak {U}(\mathbf{B})\) is a bounded linear map. In fact its operator norm is 1.

It’s obviously linear; the remaining thing to prove is that it’s bounded. The easiest manner to prove this is to think of the action of \(L\) as passive, i.e. \(\mathbf{B}\) doesn’t move under the action of \(L\); only the coordinates used to describe \(\mathbf{B}\) change. In thinking about \(L\) as a passive transformation, it’s obvious its norm is 1.

Explicitly, the norm of an element in its domain is equal to the norm of the image of that element. Hence, the ratio that appears in the definition of the operator norm

\begin{align} \| L\| \equiv \sup \limits _{\| a\| \ne 0} \frac{\| a^L\| }{\| a\| } = \sup \limits _{\| a\| \ne 0} \frac{\| a\| }{\| a\| } = 1 \end{align}

is identically 1.

With all of these ingredients prepared we can apply the Bounded Linear Transformation Theorem to extend the action of the inhomogeneous Lorentz group’s identity component uniquely to all of the quasilocal algebra \(\mathfrak {U}\), completing our examination of Axiom 5 (Lorentz Covariance).