LETTERE AL NUOVO CIMENTO VOL. 2, N. 14 2 0 t tobre 1971

The Decision Problem for Mathematical Structures of Quantum Theory.

A. BENEDETTI and G. TEPPATI

Is t i t~ to di l~isica dell' Universi t~ - T~ecce

(riccvuto il 27 Luglio 1971)

The purpose of this letter is to enunciate some results concerning the ~!ecision problem relative to a well-known mathematical structure of quantum theory. Such a structure, originally proposed by BIRKHOFF and yon NEU~ANN (1) and later devel- oped by PI toN (3) and EMCH (3) together with JAVCH (4,5), is essentially the structure of the <( propositions ~ (*) on a given quantum system and turns out to be the same of a complete, atomic, orthocomplemented, weakly modular, satisfying-the-covering-law lattice ((( system of propositions ~, in the nomenclature introduced by PIRON). Let us denote with TQ~ the theory of the above systems of propositions ; we will prove that TQM is undecidable, in a sense that will be specified below.

To begin with, it is useful to recall some definitions which are of common use in mathematical logic; they arc essentially due to TARSKI (6). The theories discussed in the following are all formalized within the lower predicate calculus, and they have the same logical constants (i.e. quantifiers, connectives, symbol of equivalence) and rules of inference. Let us call T anyone of these theories; we assume further tha t T is specified by its nonlogical constants (i.e. symbols of operations, and so on) and non- logical axioms (i.e. relations). We suppose known to the reader the definitions of a for- mula, a sentence (i.e. a formula without free variables), a provable sentence (in terms of nonlogical constants and nonlogical axioms), together with the definitions of con- sistency, completeness, and finite axiomatizabili ty of a theory T. Moreover, we suppose also known the concept of a general recursive set.

De/ in i t ion 1. - A theory T~ is an extension of a theory T 2 if every sentence provable in T 2 is also provable in T~; T, is a ]inite extension of T 2 if only finitely many axioms of T, are not provable in T 2. T 1 and T~ are said to be compatible if they have the same nonlogical constants and a consistent common extension. At last, T 1 is said to be con- sistently interpretable in T~ if T, and T 2 have a common consistent extension T such that for every nonlogical constant C in T~ which is not in T 2 there is a provable sentence in T which is a possible definition of C in terms of nonlogical constants of T 2 and possibly nonlogical constants of T.

(~) G. BIRKttOFF a~ld J . )'ON NEUMANN: Ann. Math., 37, 823 (1936). (2) C. P i tON: Helv. Phys. Acta, 37, 439 (1964). (8) C. PIRON a n d G. EMCH: Journ. Math. Phys., 4, 469 (1963). (4) J . M. JAUCH and C. PIRON: Helv. Phys..Acta, 42, 842 (1969). (5) J . M. JAUCH: Foundations o! Quantum Mechanics (New York, 1968). (*) I.e. the equiva lence classes of yes-no expe r imen t s on phys ica l sys t ems ; see on th i s purpose ref. ('). (e) A. TARSKX: Journ. Symb. Logic, 14, 75 (1949).

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6 9 6 A. BENEDETTI a n d G. T~PrAT~

The following definition points out in what a sense we speak of decidable or un- decidable theories.

Definition 2. - A theory T is decidable if the set of its provable sentences is generally recursive, and otherwise it is undecidable; T is essentially undeeidable if i t is consistent and no consistent extension of T is decidable.

The following theorems are known (*):

Theorem 1. - If a theory T is undecidable, then every theory T 1 with the same constants, of which T is a finite extension, is undccidable.

Theorem 2. - If a theory T is essentially undecidable, finitely axiomatizable and consistently interpretable in T 1, then T 1 is undecidable.

By sistematically using the preceding two theorems, it is possible to prove the next theorem.

Theorem 3. - Let T L be the theory of arbitrary lattices (formalized within the lower predicate calculus), then T L is undecidable. Moreover, let TvG be the theory of complete, complemented, modular, atomic, (h-continuous lattices (projective geometries) ; then TpG is undecidable (**).

None of the theories considered in Theorem 3 is essentially undeeidable, as TARSKI has proved that the theory Tn of arbitrary Boolean lattices is decidable.

At this point we can observe tha t the theory ~PG of complete, atomic, orthocom- plcmented, modular lattices is also undecidable (***), as a lattice of such a type is a direct union of finite-dimensional projective geometries (***). As every provable sentence in Tq~ is also a provable sentence in TPG (as in particular every complete, atomic, ortho- complemented, modular lattice is a system of propositions), ~'PG is a finite extension of TQ~, in the sense of Definition 1 (***). We are now in a condition to satisfy the requirements of Theorem 1 ; in fact, we are left with the undecidable theory Tva, which is a finite extension of the theory TQ~, so that the theory TQ~ turns out to be un- decidable. We have thus proved the next theorem.

Theorem 4. - Let Tq~ be the theory of complete, atomic, orthocomplementcd, weakly modular, satisfying-the-covering-law lattices (formalized within the lower predicate calculus); then TQ~ is undceidable.

TQ~ is no t essentially undecidable for the above reason tha t the theory T B, which is an extension of Tq~, is decidable.

Now, for what concerns physical theories, we can add the following definition:

Definition 3. - Let a physical theory T F be specified by the mathematical structure M of the set of its ~ propositions ~, and let T~ be the theory of this structure; we say tha t T F is essentially undecidable, undecidable, or decidable according to the corresponding solution given for the decision problem for T~.

Then, in the sense of Definition 3, we conclude that the theory of quantum physical systems is undecidable; at the same extent, we conclude that the theory of classical physical systems is decidable, as such a physical theory is specified by the Boolean lattice of the Borel subsets of a given phase-space.

(') See ref. (~), p. 75. (**) See ref. (~), p. 77. (*~ Obviously if not Boolean. (***) See ref. (2), theorem V. ('.*) Furthermore, it is possible to embed a system of propositions in a projective geometry in such a way that this embedding, when restricted to the points, is one-to-one, the partial order is preserved, the image of the intersection is the intersection of the images, and the image of finite unions of points is the union of the images; see ref. (1). theorem XVIII.