Quantum Mechanics, Formalization and the Cosmological Constant Problem

Abstract

Based on formal arguments from Zermelo–Fraenkel set theory we develop the environment for explaining and resolving certain fundamental problems in physics. By these formal tools we show that any quantum system defined by an infinite dimensional Hilbert space of states interferes with the spacetime structure M. M and the quantum system both gain additional degrees of freedom, given by models of Zermelo–Fraenkel set theory. In particular, M develops the ground state where classical gravity vanishes. Quantum mechanics distinguishes set-theoretic random forcing such that M and gravitational degrees of freedom are parameterized by extended real line. The large scale smooth geometry compatible with the forcing extensions is one of exotic smoothness structures of ${\mathbb {R}}^4$. The amoeba forcing makes the old real line to have Lebesgue measure zero in the extended one. We apply the entire procedure to the cosmological constant problem, especially to discard the zero-modes contributions to the gravitational vacuum density. Moreover, there exists certain exotic smooth ${\mathbb {R}}^4$ from which one determines the realistic, agreeing with observation, small value of the vacuum energy density.

Appendices

Appendix 1: The Lattice of Projections

For ${\mathcal {H}}$—a complex separable Hilbert space, let $L({\mathcal {H}})$ be the set of all projections P on ${\mathcal {H}}$, i.e. $P\cdot P=P$. Every P is a self-adjoint operator on ${\mathcal {H}}$. The range of any projection P is a unique closed subspace of ${\mathcal {H}}$. For arbitrary projections $P_1$ and $P_2$ one defines meet $P_1\wedge P_2:=P_1\cap P_2$ and join $P_1\vee P_2:=\text {span}(P_1\cup P_2)$, where span is the linear span of the sum of the ranges of $P_1$ and $P_2$. We define the natural partial order on the set $L({\mathcal {H}})$ by the condition $P_1\leqslant P_2\iff \text {ran}(P_1)\subset \text {ran}(P_2)$. Thus, the partial order $(L({\mathcal {H}}),\leqslant )$ together with $\wedge$ and $\vee$ as above is an atomic complete non-distributive lattice, called a lattice of projections. It is denoted by ${\mathbf {L}} ({\mathcal {H}})$ or simply ${\mathbf {L}}$.

The lattice ${\mathbf {L}}$ carries a logical structure of quantum mechanics. Any orthogonal projection $P:{\mathcal {H}}\rightarrow {\mathcal {H}}$ represents a proposition P in the following way. For a physical system with states represented by elements of ${\mathcal {H}}$ the proposition P corresponds to the measurement with two possible outcomes—0 and 1 depending on the fact that the state belongs to the range of P or not. One can think of such measurement as the yes–no answers to the question ‘$is\ P\ equal\ 1?$’. We say that two propositions $P_1$ and $P_2$ are orthogonal when all vectors in (the range of) $P_1$ are orthogonal to all vectors in (the range of) $P_2$, i.e. ran($P_1$)$\bot$ran($P_2$). The complement $\lnot P$ of a proposition P is defined as $\lnot P:=\text { ran} (\lnot P)=\{ x\in {\mathcal {H}}:x\bot y\text { for all } y\in \text {ran}P\}$.

Let C be an arbitrary set of commuting propositions closed under the operations of complementation and meet. An algebraic structure $(C, \vee , \wedge , \lnot , {0}, {1})$ is a propositional algebra (PA). By definition, every PA is a Boolean algebra. If we choose from the lattice ${\mathbf {L}}$ a maximal, complete Boolean algebra of projections (propositions), then for $\dim {\mathcal {H}}=\infty$ the algebra is isomorphic to the one containing the atomless part given by measure Lebesgue algebra of the interval [0, 1]. The atomless part is the measure algebra B and it is extensively used in the paper. The reason is that B generates nontrivial random forcing.

Appendix 2: Remarks on Forcing, Models of ZFC and Physics

As it was already remarked in Sect. 2 forcing in set theory was invented originally by Paul Cohen as a tool to prove the independence of the axiom of choice (AC) and the continuum hypothesis (CH) from the axioms of ZF and ZFC correspondingly. The idea is to construct models of ZFC, where AC (or CH) hold and some other models of ZF (and ZFC) where they do not. If such models exist, which indeed is true relative to the existence of the model V, then AC and CH are independent of ZF and ZFC respectively.

Soon after its invention the forcing procedure was reformulated as the property of ultrafilters on partial orders (PO) and on Boolean algebras containing these PO’s. In the 1960s of the twentieth century Scott and Solovay, and independently Vopěnka and Hajék, gave the description of forcing in terms of Boolean valued models of ZFC. The approach of this paper is based rather on the appearance of the generic filters in B in the model V, than on logical consequences of the forcing. However, there are also important logical and measure theoretic properties of the random forcing referred to (e.g. Remark 13, Theorem 3).

A forcing extension $V[{\mathcal{U}}]$ of a model V is nontrivial whenever $V\subsetneq V[{\mathcal{U}}]$ and ${\mathcal{U}}\notin V$ and ${\mathcal{U}}\in V[{\mathcal{U}}]$. The crucial for $V[{\mathcal{U}}]$ to be nontrivial is the existence of a generic filter ${\mathcal{U}}$ on a partial order $({\mathbb {P}},<)$ or on the Boolean algebra ${B}({\mathbb {P}})$ in V (see Definition 1).

Here we are going to explain how genericity of filters leads to generic extensions without refering to Boolean-valued models, i.e. this is the classical forcing description. The relation to Boolean-valued models will also be commented.

The following lemma is the analogue of Lemma 2

Lemma 11

(Jech 2003, Lemma 14.4) If $\mathcal {D}$ is a countable family of dense subsets of partially ordered set ${\mathbb {P}}$, then for every $p\in {\mathbb {P}}$ there exists a generic filter ${\mathcal{U}}$ on ${\mathbb {P}}$ such that $p\in {\mathcal{U}}$.

Then the following theorem is fundamental regarding forcing extensions

Theorem 4

(Generic model theorem, Jech 2003, Theorem 14.5) Let ${\mathbb {P}}$ be a PO in a transitive ZFC model V, and ${\mathcal{U}}\subset {\mathbb {P}}$ a generic filter on ${\mathbb {P}}$. Then there exists a transitive model $V[{\mathcal{U}}]$ of ZFC (a generic extension of V) such that:

• (1) $V\subset V[{\mathcal{U}}]$ and ${\mathcal{U}}\in V[{\mathcal{U}}]$;

• (2) The ordinal numbers in V are the same as in $V[{\mathcal{U}}]$.

Every $x\in V$ has the canonical name in $V^{B}$, denoted by $\overline{x}\in V^{B}$, such that it is the function $\overline{x}:\{\overline{y}:y\in x \}\rightarrow {B}$ for which $\overline{x}(\overline{y})=1$ for all $y\in x$. An ultrafilter $[{\mathcal{U}}]$ on B over V has the canonical name:

$$\begin{aligned} \overline{{\mathcal{U}}}:\{\overline{u}:u\in {B} \}\rightarrow {B} \text { such that } \overline{{\mathcal{U}}}(\overline{u})=u. \end{aligned}$$

(9)

For a transitive model V of ZFC and a complete Boolean algebra B in V, let ${\mathcal{U}}$ be a V-generic ultrafilter on B. Then the interpretation by ${\mathcal{U}}$ of any $x\in V^{B}$ (recall that $x\in V^{B}_{\alpha + 1}$ is a function on $V^{B}_{\alpha }$ with values in B, where $\alpha$ is a nonlimit ordinal), denoted by $x^{{\mathcal{U}}}$, is defined inductively according to:

• (1) $\emptyset ^{{\mathcal{U}}}=\emptyset$;

• (2) $x^{{\mathcal{U}}} = \{ y^{{\mathcal{U}}}:x(y)\in {\mathcal{U}} \}$.

Now, let $V[{\mathcal{U}}]$ be the transitive model of ZFC built by the ${\mathcal{U}}$-interpretations:

$$\begin{aligned} V[{\mathcal{U}}]=\{x^{{\mathcal{U}}}:x\in V^{{B}} \}. \end{aligned}$$

(10)

Let $\overline{{\mathcal{U}}}$ be the canonical name of a generic ultrafilter ${\mathcal{U}}$. Then from (9) its interpretation is $\overline{{\mathcal{U}}}^{{\mathcal{U}}}={\mathcal{U}}$. Hence, from (10), it follows that ${\mathcal{U}}\in V[{\mathcal{U}}]$.

Finally, note that the model $V[{\mathcal{U}}]$ is isomorphic to the quotient $V^{{B}}/_{{\mathcal{U}}}$ (Jech 2003, p. 224, Ex. 14.15). This last property lies at heart of the Boolean-valued approach to forcing and was crucial in Sect. 2.

Note that from physics point of view the Boolean valued model $V^B$ contains all information about generic extensions $V[{\mathcal{U}}_{\alpha }]$ for all $\alpha \in I$, so $V^B$ is a kind of (Boolean) interference of the extensions. Forcing is tightly connected to the local hidden variable program in QM and to the spectral theorem (see e.g. Klimasara and Król 2015; Van Wesep 2006; Takeuti 1978). In the LHV program one can ask the question about the conditions which must be fulfiled by the formalism of QM in order a semiclassical state $\psi _{\mathrm{cl}}$ exist. Such state would make possible a simultaneous measurement of arbitrary observables giving definite results. $\psi _{\mathrm{cl}}$ does not exist in a single model V, however it does in the generic extension $V[{\mathcal{U}}]$ as a generic ultrafilter. It depends on ${\mathcal{U}}$ such that there is no single model containing all forcing extensions (otherwise this would realize LHV in the model). This is yet another reason for the family of forcing extensions of V would not be a single-element (cf. Lemmas 4, 5 and Corollary 2). The following theorem explains this important phenomenon

Theorem 5

(Klimasara and Król 2015) For any complete atomless Boolean algebra B (of spectral measures of quantum observables), any semiclassical state $\psi _\mathrm{cl}$ is well defined generic filter ${\mathcal{U}}$ in B. Such a classical state is added by the nontrivial forcing to the ground Boolean model $V^{B}$ of ZFC, i.e. ${\mathcal{U}}\in V^{B}[{\mathcal{U}}]$ and ${\mathcal{U}}\notin V$ (${\mathcal{U}}$ is the generic ultrafilter on B in V).