On Particle Phenomenology Without Particle Ontology: How Much Local Is Almost Local?

Abstract

Recently, Clifton and Halvorson have tried to salvage a particle phenomenology in the absence of particle ontology within algebraic relativistic quantum field theory. Their idea is that the detection of a particle is the measurement of a local observable which simulates the measurement of an almost local observable that annihilates the vacuum. In this note, we argue that the measurements local particle detections are supposed to simulate probe radically holistic aspects of relativistic quantum fields. We prove that in an axiomatic (Haag-Araki) quantum field theory on Minkowski spacetime, formulated in a Hilbert space $\mathcal{H}$, there is no positive observable C, with norm less than or equal to 1, satisfying the conditions: (1) the expectation value of C in the vacuum state Ω is zero, (2) there is at least one vector state Ψ in $\mathcal{H}$ in which the expectation value of C is different from zero, and (3) there exists at least one spacetime region $\mathcal{O}$ such that the non-selective measurement of C leaves the expectation values of all observables in the local algebra of that region unaltered regardless of the state the system is in. The result reveals a tension between intuitions regarding localization and intuitions regarding causality: to save “particle phenomena” in the absence of particle ontology, one has to feign “particle” detectors with “good” properties as to locality but “bad” behavior as to causality.

Appendices

Appendix 1: Positive Almost Local Operators that Annihilate the vacuum

To begin with, an operator $L \in \mathcal{R}$ is said to be almost local just in case there exists a net $\mathbb{R}^{ +} \ni r \mapsto L_{r} \in \mathcal{R}( \mathcal{O}_{r})$ of local operators associated with the double cones (“diamond regions”) $\mathcal{O}_{r} = \{ x \in\mathbb{R}^{4}:\vert x^{0} \vert + \vert \vec{x} \vert < r\}$ in Minkowski spacetime [endowed with global inertial coordinates $(x^{0},\vec{x})$] such that for every n=1,2,…

$$ \lim_{r \to\infty} r^{n}\Vert L - L_{r} \Vert = 0. $$

(9)

It follows that for any prescribed (arbitrarily small) inaccuracy δ>0, there exists a radius r=r(δ) (whose value depends on δ) such that the local algebra pertaining to the double cone $\mathcal{O} _{r(\delta)}$ with that radius contains a local operator L′ within norm distance δ from the almost local operator L:

$$ \bigl \Vert L - L' \bigr \Vert < \delta \quad\mbox{for some}\ L' \in \mathcal{R}(\mathcal{O}_{r(\delta)}). $$

(10)

Such almost local operators are constructed by smearing the spacetime translates of local operators with appropriate test functions of rapid decrease. Specifically, take some local operator A associated with some open bounded region $\mathcal{O}$, $A\in \mathcal{R}(\mathcal{O})$, and define the operator

$$ L\mathop{ =} \limits_{\mathrm{def}} A(f) = \int d^{4}x\,f(x)U(x)AU(x)^{*} $$

(11)

where f belongs to the Schwarz space $\mathcal{S}(\mathbb{R}^{4};\mathbb{C})$, i.e., the space of complex infinitely differentiable functions on $\mathbb{R}^{4}$ which, together with their derivatives, approach zero at infinity faster than the inverse of any power of the Euclidean distance. The operator L=A(f) defined via (11) is provably an almost local operator. Smearing appropriately the spacetime translates of the original local operator A with test functions on $\mathbb{R}^{4}$ of compact support effects the approximation in the norm of the almost local operator L=A(f) by local operators.

Moreover, suppose f arises from the Fourier transform of a function $\tilde{f}$ with compact support Δ in the “energy-momentum space”. If Δ lies outside the closed forward light cone $\bar{V}^{ +}$, then the almost local operator L=A(f) annihilates the vacuum,

$$ L\varOmega= A(f)\varOmega= 0. $$

(12)

Finally, to get a positive almost local operator that annihilates the vacuum, one can just set C=L ^∗ L=A(f)^∗ A(f).

Thus one can construct positive almost local operators that annihilate the vacuum but are approximated arbitrarily close in the norm by local operators associated with bounded regions of Minkowski spacetime. So, all the elements necessary to implement the Halvorson-Clifton account of particle phenomenology with no particle ontology, discussed in Sect. 2, are in place.

Appendix 2: Proof of Lemma

Let C be an effect on a Hilbert space $\mathcal{H}$ such that for every self-adjoint operator A in a concrete C ^∗-algebra $\mathfrak{A}$, A=T _C (A), where T _C (A) is defined as in (6) above. We shall show that C ^1/2 commutes with every element of $\mathfrak{A}$.

Consider an arbitrary self-adjoint operator $A \in\mathfrak{A}$. Since A=T _C (A),

$$ \bigl( T_{C}(A) - A \bigr)^{*} \bigl( T_{C}(A) - A \bigr) = 0. $$

(13)

Given that C is an effect and A is self-adjoint, (13) implies that

$$ T_{C}(A)^{2} - T_{C}(A)A - AT_{C}(A) + A^{2} = 0. $$

(14)

Next, via an elementary calculation that deploys (6) together with the identities A ²=T _C (A)² and A ²=T _C (A ²) (A ² is also a self-adjoint operator in $\mathfrak{A}$), one may arrive from (14) to

$$ \bigl[ A,C^{1/2} \bigr]^{*} \bigl[ A,C^{1/2} \bigr] + \bigl[ A,(I - C)^{1/2} \bigr]^{*} \bigl[ A,(I - C)^{1/2} \bigr] = 0. $$

(15)

The left-hand side of (15) is a sum of non-negative elements of the C ^∗-algebra $\mathcal{B}(\mathcal{H})$ of all bounded linear operators on $\mathcal{H}$. Hence (15) leads to

$$ \bigl[ A,C^{1/2} \bigr] = 0. $$

(16)

Therefore (16) holds for every self-adjoint element $A \in\mathfrak {A}$. And since every element of $\mathfrak{A}$ can be written as a sum of two self-adjoint elements of $\mathfrak{A}$, one may conclude that

$$ \bigl[ C^{1 / 2},F \bigr] = 0\quad \forall F \in\mathfrak{A}. $$

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