Elemental principles of t-topos

– In this paper, a sheaf-theoretic approach toward fundamental problems in quantum physics is made. For example, the particle-wave duality depends upon whether or not a presheaf is evaluated at a specified object. The t-topos theoretic interpretations of doubleslit interference, uncertainty principle(s), and the EPR-type non-locality are given. As will be explained, there are more than one type of uncertainty principle: the absolute uncertainty principle coming from the direct limit object corresponding to the refinements of coverings, the uncertainty coming from a micromorphism of shortest observable states, and the uncertainty of the observation image. A sheaf theoretic approach for quantum gravity has been made by Isham-Butterfield in (Found. Phys., 30 (2000) 1707), and by Raptis based on abstract differential geometry in Mallios A. and Raptis I., Int. J. Theor. Phys., 41 (2002), qr-qc/0110033; Mallios A., Remarks on “singularities” (2002) qr-qc/0202028; Mallios A. and Raptis I., Int. J. Theor. Phys., 42 (2003) 1479, qr-qc/0209048. See also the preprint by Requardt M., The translocal depth-structure of space-time, Connes’ “Points, Speaking to Each Other”, and the (complex) structure of quantum theory, for another approach relevant to ours. Special axioms of t-topos formulation are: i) the usual linear-time concept is interpreted as the image of the presheaf (associated with time) evaluated at an object of a t-site (i.e., a category with a Grothendieck topology). And an object of this t-site, which is said to be a generalized time period, may be regarded as a hidden variable and ii) every object (in a particle ur-state) of microcosm (or of macrocosm) is regarded as the microcosm (or macrocosm) component of a product category for a presheaf evaluated at an object in the t-site. The fundamental category Ŝ is defined as the category of ∏ α∈∆ Cα-valued presheaves on the t-site S, where ∆ is an index set. The study of topological properties of S with respect to the nature of multi-valued presheaves is left for future study on the t-topos version of relativity (see Kato G., On t.g. Principles of relativistic t-topos, in preparation; Guts A. K. and Grinkevich E. B., Toposes in General Theory of Relativity (1996), arXiv:gr-qc/9610073, 31). We let C1 and C2 be microcosm and macrocosm discrete categories, respectively, in what will follow. For further development see also Kato G., Presheafification of Matter, Space and Time, International Workshop on Topos and Theoretical Physics, July 2003, Imperial College (invited talk, 2003). Basic definitions. – For t-topos theory, the notion of a t-site plays the role of hidden variables. More conditions will be added to the site when the further applications in [1] are made. For the concept of a Grothendieck topology, see [2–4] or [5]. Definition 1.1 . Let S be a site, namely, a category with a Grothendieck topology and let Ŝ be the category of presheaves from S to the product category ∏ α∈∆ Cα. That is, Ŝ = ( ∏ α∈∆ Cα) Sopp , where S is the dual category of S. Then site S is said to be a temporal site or simply t-site when S is used in this context. Category Ŝ is said to be a t-topos or temporal topos. We sometimes call an object of Ŝ an entity. Remarks 1.2 . i) See [3] or [5] for Grothendieck topologies which is sufficient for t-topos theory. ii) For an object F in Ŝ, which we write as F ∈ Ob(Ŝ) and for an object V in S, i.e., V ∈ Ob(S), F (V ) is an object in α∈∆ Cα. Namely, F (V ) = (F (V )α)α∈∆ , where F (V )α is the α-th component of F (V ). We also say that F (V ) is the manifestation of F at the generalized time period V . Definition 1.3 . Let F be an object of Ŝ. The state of F during a generalized time period W , namely, an object of S, is defined by the pair (F,W ) = F (W ). Then F is said to be manifested during W . When a generalized time period is not given, F is said to be in a pre-state or in an unmanifested state. (See Note 1.4′ below.) For a specified object V , the object F (V ) is said to be in the particle ur-state of F over the generalized time period V , and when one object in the t-site is not specified for F , then F is said to be in a wave ur-state of F and sometimes denoted as {F (W )}W∈Ob(S) or F (−). Definition 1.4 . An observation of an object m of Ŝ by another object P of Ŝ in a nondiscrete category Cα, α ∈ ∆, over a generalized time period V is a natural transformation s over V . Namely, the morphism in Cα sV : m(V ) −→ P (V ) (1) is said to be an observation of m by P during the generalized time period V . If such a natural transformation s over a specified object V of t-site exists, then m is said to be observable or measurable by P during the generalized time period V . We may also say that m interacts with P if there exists such a natural transformation from m to P over some generalized time period. Notice that when m is measured, m needs to be in a particle ur-state since an object in S must be specified for the natural transformation in (1). Note 1.4′. When an object m of Ŝ is not observed, not only m is in the wave ur-state, i.e., {m(V )} in Definition 1.3, but also (we will be more precise in Definitions 2.1 and 2.2) m may be considered as the totality of decomposed subobjects of m which are to be evaluated at unspecified objects of S. It may be most appropriate to consider an unobserved object m to be simply presheaf “m”. Note 1.5 . Let {Vi → V } be a covering of V and let {Vi←j → Vi} be a covering of Vi as in [2–6] or [7]. Then by composing covering morphisms, {Vi←j → V } is a covering of V . Similarly, by composing further, one gets a covering {Vk←j←i → V } of V . Then, consider the inverse limit covering { lim ← V...←k←j←i −→ V } (2) of V . In the next section, we will need this notion. Definition 1.6 . Let C1 be the microcosm discrete category. That is, an object of C1 is a particle in microcosm, and as a category, C1 is discrete, namely, no morphisms exist except identity morphisms. Note 1.7 . The topos approach in [8] and [9] by Butterfield-Isham can be interpreted in terms of t-topos as follows. First, we will explain the basic method in [8] and [9]: Let S be the state space and let A be a physical quantity and let Ā be a real-valued function representing A as in [8]. Then the functional composition principle (referred to as FUNC in [8] and [9]) is the commutative diagram