A Topos Foundation for Theories of Physics: III. The Representation of Physical Quantities With Arrows

摘要

This paper is the third in a series whose goal is to develop a fundamentallynew way of viewing theories of physics. Our basic contention is thatconstructing a theory of physics is equivalent to finding a representation in atopos of a certain formal language that is attached to the system. In paper II,we studied the topos representations of the propositional language PL(S) forthe case of quantum theory, and in the present paper we do the same thing forthe, more extensive, local language L(S). One of the main achievements is tofind a topos representation for self-adjoint operators. This involves showingthat, for any physical quantity A, there is an arrow$\breve{\delta}^o(A):\Sig\map\SR$, where $\SR$ is the quantity-value object forthis theory. The construction of $\breve{\delta}^o(A)$ is an extension of thedaseinisation of projection operators that was discussed in paper II. Theobject $\SR$ is a monoid-object only in the topos, $\tau_\phi$, of the theory,and to enhance the applicability of the formalism, we apply to $\SR$ a toposanalogue of the Grothendieck extension of a monoid to a group. The resultingobject, $\kSR$, is an abelian group-object in $\tau_\phi$. We also discussanother candidate, $\PR{\mathR}$, for the quantity-value object. In thispresheaf, both inner and outer daseinisation are used in a symmetric way.Finally, there is a brief discussion of the role of unitary operators in thequantum topos scheme.