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A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory

### Abstract

This paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper, we study in depth the topos representation of the propositional language, , for the case of quantum theory. In doing so, we make a direct link with, and clarify, the earlier work on applying topos theory to quantum physics. The key step is a process we term “daseinisation” by which a projection operator is mapped to a subobject of the spectral presheaf—the topos quantum analog of a classical state space. In the second part of the paper, we change gear with the introduction of the more sophisticated local language . From this point forward, throughout the rest of the series of papers, our attention will be devoted almost entirely to this language. In the present paper, we use to study “truth objects” in the topos. These are objects in the topos that play the role of states: a necessary development as the spectral presheaf has no global elements, and hence, there are no micro-states in the sense of classical physics. Truth objects therefore play a crucial role in our formalism.

© 2008 American Institute of Physics

Received 21 May 2007
Accepted 25 January 2008
Published online 13 May 2008

Acknowledgments:
This research was supported by grant RFP1-06-04 from The Foundational Questions Institute (fqxi.org). A.D. gratefully acknowledges financial support from the DAAD. This work is also supported in part by the EC Marie Curie Research and Training Network European Network on Random GEometry (ENRAGE) MRTN-CT-2004-005616. We are both very grateful to Professor Hans de Groote for his detailed and insightful comments on our work.

Article outline:

I. INTRODUCTION
II. QUANTUM PROPOSITIONS AS SUBOBJECTS OF THE SPECTRAL PRESHEAF
A. From projections to global sections of the outer presheaf
1. The definition of
2. Properties of the mapping
3. A logical structure for ?
4. Hyper-elements of
B. Daseinisation
1. From global sections of to subobjects of
2. Heidegger encounters physics
C. The Heyting algebra structure on
D. Daseinisation and the operations of quantum logic
1. The status of the possible axiom
2. Inner daseinisation and
III. THE SPECIAL NATURE OF DASEINISED PROJECTIONS
A. Daseinised projections as optimal subobjects
IV. TRUTH VALUES IN TOPOS PHYSICS
A. The mathematical proposition
B. Truth objects
1. Linguistic aspects of truth objects
2. Truth objects in a general topos
3. The example of classical physics
4. The truth object in quantum theory
5. Time dependence and the truth object
C. The presheaf
1. The definition of
2. The monic arrow from to
V. CONCLUSION

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2008-05-13

2016-08-24

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