Quantum Structures and Causation Brussels, 30 Nov.- 1 Dec. 2013 - - PDF document

Whither Quantum Structures?

Quantum Structures and Causation

Brussels, 30 Nov.- 1 Dec. 2013

Karin Verelst

CLEA, Vrije Universiteit Brussel kverelst@vub.ac.be

Summary

• What can quantum logic do for philosophy?
• What can cateogry theory do for philosophy?
• Causation
• Identity and Causation
• Order and Causation
• Noether’s Theorem

What can quantum logic do for Philosophy? Birkhoff and Von Neumann (1936): compare the logical structures of physical theories rather than their specific contents in order to understand their differences. Two levels of description for the characterisation of phys- ical systems in general, each equipped with an appro- priate space and a notion of proposition:

• bservational level
• observation space := outcome sets for given exper-

iments

• experimental propositions := certain subsets of the
• utcome sets which represent statements on the

results of specific measurements, to each of which a probability is assigned

state level

• state space
• propositions := certain subsets of the state space

which are related to specific subsets of the out- come sets The problem is to characterise this relation both in the classical and the quantum case. Link to logic: notion of experimental propositions as an- swers to yes/no-questions. Ex.: “does the temperature

• f the gas lie within a given interval of R+?”).

In the quantum case the yes/no-questions correspond to projection operators P, i.e., self-adjoint, idempotent (P 2 = P) operators with as spectrum the set {0, 1}, which are in one-to-one correspondence with the closed

• nedimensional subspaces of H .

BVN saw that the connection between observables and projections paves the way for a logical perspective on experimental propositions via the calculus of projec-

• tions. Their concern with logic arose from the fact that

the propositions on the QM-observables of a physical

system do not behave in accordance with the laws of classical logic: distributivity fails. Ontological interpretation of this failure: nature of “state” (superpositions), nature of “system” (whole greater than sum parts), nature of “‘measurement” (incompatible mea- surements)! Remark: Experimental propositions depend a priori only

• n the measurement apparatus although, in order to be

able to set up an experiment, one has to make implicit assumptions on the nature of the system observed. Mackey (1963): lattice theoretical formulation of the cal- culus of projections. Notion of “property lattice”. Fundamental contribution by Piron:

• “operational” approach: a set of axioms for QL based
• n physical considerations;
• representation theorem (1964, 1972, 1976), connect-

ing the property lattice of the OQL-description to Hilbert space

• rigourous definition of “experimental project”: distinc-

tion between states, properties and measurement out- comes (“elements of reality” counterfactually defined)

Problems:

• only pure states
• no lattice-theoretic representation for the tensor prod-

uct between Hilbert spaces (compound systems) But we did learn a lot! “(...) it must above all be recognized that, however far quantum effects transcend the scope of classical phys- ical analysis, the account of the experimental arrange- ment and the record of the observations must always be expressed in common language supplemented with the terminology of classical physics.” (N. Bohr, 1948) “(...) pushing this system [a measuring instrument] into the framework of quantum mechanics, one should ad- mit that it is never possible to superpose states cor- responding to different positions of the pointer, for this would contradict the very assumption that one has to do with a measuring instrument. Again, we say that states

• f such a system are separated by a superselection
• rule. Of course, when a superselection rule operates

between any two states, thus dismissing entirely super- positions of states, one loses quantum behavior com- pletely and reverts to classical behavior.” (Beltrametti and Casinelli, 1981) Experiments as interventions

• Isolation of a part of the world
• Preparation of the physical system obtained
• Classical reduction: actual measurements

Gedanken-example: the Puree-experiment (Verelst, IQSA 2002; IAFE, 2011) : a blindfolded observer digs in the ground and finds a potatoe. In order to find out its prop- erties he subjects it to various treatments (I guide him and hand him the instruments he needs). (‘I’ ≡ suppo- sitions).

✛ ✚ ✘ ✙

Si (Potatoe) − → Sf (mashed potatoes, puree)

take it, peel it, cook it, and mash it

1) The intervention changes the original system. 2) Its result is such that it can be expressed by means

• f finitely many logical propositions (yes/no-questions).

3) The final step is in general irreversible! HOWEVER: received experimental practice consists (af- ter standardised repitition) in concluding that Sf = Si, i.e., that this part of the world consists of puree!

1

• π(l′)
• π(r′)
• π(n′)
• π(f ′)
• π(b′)
• π(l)
• π(r)
• π(n)
• π(f)
• π(b)
• The logic of Firefly experimental propositions

1

• {r′}
• {b′}
• {n′}
• {f ′}
• {l′}
• {l}

{r′b′} {f} {r′f ′} {l′b′} {b} {l′f ′} {r} {lf}

• {lb}
• {n}
• {rf}
• {rb}
• The logic of Firefly properties

.... both Π(A) [the experimental propositions] and L [the physical properties] are isomorphic to the lattice of closed subspaces of the Hilbert Space. Although this is mathematically quite convenient, it has encouraged people to identify Π(A) with L and this has been a metaphysical disaster. ...” (C.H. Randall and D.J Foulis, 1983) .... The affirmations of the phycicist in regard to a physi- cal system are susceptible of being regulated by exper-

• iment. This control consists in general of a measure-

ment the result of which is expressed by “yes” or “no”.” (C. Piron, 1976) .... the polemic surrounding the problem shows an ir- reducible opposition between the language of classical physics and quantum physics. This oppostion is partic- ularly irritating because in quantum physics, one must make an appeal to classical concepts to describe mea- surements.” (Piron 1976)

What can category theory do for Philosophy? 2005, “Impact of Categories” conference at ´ Ecole Nor- male Sup´ erieure in Paris: Categories are a subject full of possibilities for other fields, especially philosophy. But surprisingly little work is done in philosophy using category theory (CT), either as a formal or conceptual tool. This situation has not much changed! E.g., contemporary structuralists talk “in the air” about isomorphism (structural identity), while CT seems to be such a natural framework for the development of their ideas (Rodin, 2007). Why is this? In philosophy, ideas never are just “interesting” or “use- ful”; there are always (implicit) ontological assumptions, hidden in the formal structures of the theory. The case in point is the role of the PC in classical logic since Aris- toteles onwards (Lukasiewicz, 1933). This sheds some light on the strength of Set Theory (ST) as a founda- tional tool.

Quick comparison between CT and ST: Set Theory (Naive, ZFC, . . . ) Cantor: “a collection of objects thought of as a whole”∗

• > ontology of whole and parts: break an object into its

parts/look what it is constituted by (e.g., the real line) Primitive “element” relationship: a ∈ X ; this is a logical proposition (truth values, subject to logical operators) Intrinsic extensionality: a set is completely detemined by its elements: whole “=” parts -> but problematic: set theoretical paradoxes! (‘to be’ = ‘to have a property’) Category Theory A mathematical theory conceived to formalise structure from an external point of view: how something relates to its environment (= set theory: internal structure: “∈”). [Eilenberg and Mac Lane, 1945] A category is a collection of objects (structures) with the structure preserving relationships between them (mor- phisms)

∗“Unter einer ‘Menge’ verstehen wir jede Zusammenfassung M

von bestimmten wohlunterschiedenen Objekten m unserer An- schauung oder unseres Denkens (welche die ‘Elemente’ von M genannt werden) zu einem Ganzen (1962:282).”

A category C is defined by (Borceux, vol. I, 1994)

• a class of objects |C|,
• a class of “structure preserving” arrows f ∈ C(A, B)

called morphisms (HomC(A, B))

• an identity arrow f ◦ 1A = f; 1A ◦ g = g
• associative arrow composition: (f, g) → g ◦ f, with

h ◦ (g ◦ f) = (h ◦ g) ◦ f ∃ many examples in mathematics of structured sets and the functions between them, but also in logic, informa- tion theory, physics,... Ideal to capture the notion of types and the processes between them (B. Coecke, in: What is Category Theory, p. 47). It has very general meaning and applicability, because the conditions imposed on induced relationships are uni- versal -> proofs by means of commuting diagrams (ex.: categorical product). Interesting feature: “meta-level” arises naturally (= set theory!) Cat C:

• objects: A, B, C ∈ |C|,
• arrows: morphisms f : A → B

Cat Cat:

• objects: small categories C, D,
• arrows: functors F : C → D

Cat FunCat:

• objects: functors F, G
• arrows: natural transformations F =

⇒ G

! Cat: the category of small categories and with as mor-

phisms functors between small categories. Cat is not itself small, and therefore not an object of itself Object is a black box, its external relationships deter- mine its internal structure logically: intensional approach Categories (‘large’ and ‘small’) exits freely in hierarchies, can ‘contain’ each other. (C small: ob(C) and hom(C) are sets; C large otherwise (universal types of structure) an intrinsic connection between CT and ST by way of the Yoneda Lemma “Abstract”? Bob Coecke: “They are out there in the woods!” (Paris 2005)

⋆ types and processes ⋆ systems and measurements ⋆ data and programs ⋆ models and theories . . . Morphism: duality of structures, but without plain re- versibility (if inverse exists => isomorphism: ‘structural’ identity). Different notion of structure: coherent behaviour; change, but with respect to “something”. In physical situations, you want to make sure that the system you’re talking about ‘before’ and ‘after’ the process is the same: some- how ‘identity’ is conserved through change.

Causation Can I, as a philosopher, do something with what I learned here? How I can use QL and/or CT as a tool when deal- ing with specific philosophical problems? YES! Problem: causation. Why? ∃ many conflicting concep- tions, but they need to have something in common, be- cause otherwise no point in using the notion at all. − → “structural” approach This intuition that different “causal” theories share some common characteristic I shall try to catch categorically and see whether this brings us closer to understanding causation without a priori attempting to define it

Causation “When two events belong to one causal line (causal pro- cess) the earlier may be said to cause the later. In this way laws of the form ‘A causes B’ may preserve a cer- tain validity.” (Russell, 1948, p. 334) “I call a series of events a “causal line” if, given some of them, we can infer something about the others without having to know anything about the environment.” (1948,

• p. 333)

“A causal line may always be regarded as a persistence

• f something, a person, a table, a photon, or what not.

Throughout a given causal line, there may be constancy

• f quality, constancy of structure, or gradual changes

in either, but not sudden change of any considerable magnitude.” (1948, pp. 475-7)

• ∃ some necessary connection between (at least)

two “events” (a “cause” and an “effect”).

• asymmetrical with respect to time.

! None of the theoretical constructions granting causal-

ity can be completely reduced to the empirical level (Hume). They shed light on each theory’s hidden metaphysical assumptions, to fill out the ‘gaps’. Problem: avoid para- doxes of ‘sameness’, plurality (divisibility) and motion (change) — from Zeno to Kant! Solution? a general strategy which divides the “world” into a statical layer of stability and a dynamical one of change, connected by a relation of causality. Claim: a metaphysics is a structural scheme to which all theories that give a partial or a total description of the world comply, and in which the following dual relation- ships hold : identity

causation

• time

change

participation

• causation = forward implication;

participation = backward implication.

The backbone structure of the scheme is logical in the classical sense: grounded in the principle of contra- diction [Arist., Met., Γ,3,1005b(19-26); B 2,996b(30)]. This strategy := a classical metaphysics (CMet). The identity-level allows to classify the causally gener- ated individual instances. CMet therefore does come down to a more general global–local duality: universal

Localisation

• change

particular

Globalisation

• A metaphysical theory is not just a “theory of what there

is” (Quine). In this sense modern physics is as meta- physical as Aristotle.

The universal level connects instances which are dis- tinguished on the purely empirical, particular level. In (experimental) sciences this is expressed by the idea of physical law, universally valid on a “scene”, itself out- side of observational reach The general validity of the principle that the universe presents the same aspect from every point (...) is ac- cepted in modern physics as a necessary condition for the repeatability of experiments, since space and time are the only parameters which, at least in principle, are beyond the control of the experimenter and can not be reproduced at his will. (M. Jammer, Concepts of space,

• p. 84.)

Order and Causation The notion of “natural law” lies within the realm of the identical (the universal), and therefore does not yet itself express causality! ´ Emile Meyerson analysed — in his Identit´ e et R´ ealit´ e — the precise relationship between identity and causal- ity: notre croyance ´ a l’homog´ en´ eit´ e de l’espace implique quelque chose de plus que la persistance des lois. Nous sommes (...) convaincus que non seulement les lois, mais encore les choses elles-mˆ emes ne sont pas mod- ifi´ ees par leur d´ eplacement dans l’espace. (p. 29) In physical theories, causality is a relation between space and time, not just between objects or events: le principe de causalit´ e exige l’application au temps d’un postulat qui, sous le r´ egime de la l´ egalit´ e seule, ne s’applique qu’` a l’espace. (p. 32)

✬ ✫ ✩ ✪

Ainsi le principe de causalit´ e n’est que le principe d’identit´ e appliqu´ e ` a l’existence des objets dans le temps. (p. 37) = ⇒ principle of conservation of identity through time

Order and Causation Causality has been interpretated as an order relation

“stronger than” both logically and physically in, e.g.,

Leibnizian mechanics, information theory, “branching space-time”, quantum logic... This naturally leads to the lattice theoretical description of physical systems (Vick- ers, Topology via Logic, 1989). There are two funda- mental notions: the system’s set of states Σ and its lat- tice of properties L. A (meet, join) lattice (L, ∧, ∨) is an

• rder lattice when a < b if a ∧ b = a or a ∨ b = b.

Vice versa when a∧b =inf(a, b), and a∨b =sup(a, b). Depending on the problem at hand more structural con- ditions can be imposed (completeness, atomisticity, or- thogonality,...). Piron (in the “Green Book”) explicitly interpretes as a physical law. A property is counterfactually defined as an equivalence class of measurements that could be performed on S and then would lead to a certain result (“elements of reality”). Evidently, there is no 1- 1-relationship between states and properties (Mackey’s yes/no questions). The lattice of properties is related to the system’s set of states Σ by the notion of actual-

• ity. There is a strong connection between counterfactu-

als and causal relations! (D. Lewis (Counterfactuals), J. Pearl (Causality: Models, Reasoning, and Inference).

With any “physical” poset (S, ) we can associate two “horizontal” categories : Cat SΣ:

• objects: states ∈ Σ,
• arrows: state transitions f : P(Σ1) −

→ P(Σ2) Cat SL:

• objects: properties ∈ L,
• arrows: property transitions g : L1 −

→ L2 But there exists another map µ∗ : P(Σ) → L. It maps a subset A of Σ on the strongest (infimal) property that is true whenever a state p ∈ A is actual. The reverse map µ : L → P(Σ) maps property a on the small- est (supremal) set of all possible states p that make the property true.† In the context of lattice theory the adjoint arises natu- rally as the Galois connection (Aerts, 1994; Coecke & Stubbe, 1999) —> expression of the physical duality between properties and states!

†D.J. Moore, “On State Spaces and Property Lattices”, Stud. Hist.

• Phil. Mod. Phys., 30, 1999).

In category theory, we have the notion of naturality: a kind of coherence condition on arrows on different lev- els, a way of transforming a functor into another one while respecting the composition of morphisms in the underlying categories. Let F, G be functors from cate- gory C to category D. Assign to each A ∈ |C | a mor- phism θA ∈ Hom(FA, GA) in D, such that for any f ∈ Hom(A, B) in C : θB ◦ Ff = Gf ◦ θA. A natural trans- formation F = ⇒ G is a map θ over all components A. ∃ specific kind of natural transformation which captures behaviour and formal properties of structural relations in terms of the optimal solution locally for a problem posed globally: adjunction “Adjoint functors arise everywhere.” (Saunders Mac Lane, Categories for the working mathematician) Claim: causation in (meta)physical theories has the for- mal structure of an adjoint

Noether’s Theorem

We saw that a physical law does not embody the causal aspect of a theory. Are state-property transitions lawful

• r causal? In the order-case, counterfactuality seems to

provide for the connection between law and cause. Is on this basis a more general approach towards causation possible? Can we recover the adjoint-relationship in a differernt way? Conjecture: via Noether’s Theorem Put in a slogan, NT says that every differential symme- try of the action of a system has a corresponding con- servation law. More precisely: If the Lagrangian for a given physical system S is not affected by changes in the coordinate system used to describe it, then there will be a corresponding conservation law. This is true for physical theories in general, not only CM. The re- lation between symmetry and conservation is mediated by a minimisation principle, the Principle of Least Action (Hamilton’s Principle). According to Hamilton’s princi- ple, S, the action of the system, is invariant under con- tinuous transformations, i.e., the path taken by S be- tween t1 and t2 is the one for which the action is sta- tionary: δS = 0.

The catch is that these transformations are symmetri- cal with respect to time. Noethers theorem deals pre- cisely with this relation between symmetry and conser-

• vation. The continuous transformations can be of differ-

ent kinds:

• uniformity of time: the Lagrangian of a closed system

does not depend on time explicitly (the ‘origin’ of time is irrelevant): conservation of energy

• homogeneity of space: the properties of a closed sys-

tem do not change during a parallel displacement (the

• rigin of cinematical motion is irrelevant): conservation
• f momentum
• isotropy of space: the properties of a closed system

do not change under rotation of the system as a whole (the origin of dynamical motion is irrelevant): conserva- tion of angular momentum In the notion of conservation, the lawful and the causal aspect coincide! Moreover, Noether’s Theorem is very general: it holds for point particles (differential equa- tions), fields (partial differential equations), and gauge theory (the Standard Model). It also holds in SRT and GRT.

Question: does the specific naturality condition needed for categorical adjunction hold also in this case? How to prove? suggestion (F . Holik): exploit the con- nection between the automorphisms on the lattice (the symmetry group) and the symmetries for the Hamilta- nian transformation Ut: H − → exp

• −iHt