- ...
^{1} *cecilia.flori@aei.mpg.de*.

- ... complete
^{2} - A set
is said to be complete if all history are
pair-wise disjoint and their logical `or' forms the unit history
.

- ... formalism
^{3} - The acronym `HPO' stands
for `history projection operator' and was the name given by Isham
to his own (non-topos based) approach to consistent-history
quantum theory. This approach is distinguished by the fact that
any history proposition is represented by a projection operator in
a new Hilbert space that is the tensor product of the Hilbert
spaces at the constituent times. In the older approaches, a
history proposition is represented by a sum of products of
projection operators, and this is almost always
*not*itself a projection operator. Thus the HPO formalism is a natural framework with which to realise `temporal quantum logic'..

- ... history
^{4} - The
*unit history*is the history which is always true.

- ... theorem
^{5} -
**Kochen-Specker Theorem**: if the dimension of is greater than 2, then there does not exist any valuation function from the set of all bounded self-adjoint operators of to the reals such that for all and all , the following holds ..

- ... realist
^{6} - By a
`realist' theory we mean one in which the following conditions are
satisfied: (i) propositions form a Boolean algebra; and (ii)
propositions can always be assessed to be either true or false. As
will be delineated in the following, in the topos approach to
quantum theory both of these conditions are relaxed, leading to
what Isham and Döring called a
*neo-realist*theory..

- ... transform
^{7} - Given a commutative von Neumann
algebra V, the Gel'fand transform is a map
(25) (26)

where is the Gel'fand spectrum; is such that ..

- ...
spectrum
^{8} - Given an algebra V, the
*Gel'fand spectrum*, , is the set of all multiplicative, linear functionals, , of norm 1..

- ...
points
^{9} - In a topos
, a `point' (or `global element';
or just `element') of an object
is defined to be a morphism
from the terminal object,
, to
.
.

- ... let
^{10} - We will
denote the set of open subsets of a topological space,
, by
.
.

- ...
subsets
^{11} - Arguably, it is more appropriate to represent
propositions in classical physics with Borel subsets, not just
open ones. However, will not go into this subtlety here.
.

- ... topoi
^{12} - Of course, in the case
of temporal logic, the Hilbert spaces
and
are
isomorphic, and hence so are the associated topoi. However, their
structural roles in the temporal logic are clearly different. In fact,
in
the closely related situation of composite systems it will
generally be the case that
and
are
**not**isomorphic. Therefore, in the following, we will not exploit this particular isomorphism..

- ...
topoi
^{13} - We are here exploiting the trivial fact that, for
any pair of categories
, we have
.
.

- ... that
^{14} - Since there is no
state-vector reduction the existence of an operation
between truth values ,
that satisfies equation (100) is plausible. In fact,
unlike the normal logical connective `
', the meaning of the
temporal connective `
' implies that the propositions it
connects do not `interfere' with each other since they are
asserted at different times: it is thus a sensible first guess to
assume that their truth values are independent.
The distinction between the temporal connective ` ' and the logical connective ` ' is discussed in details in various papers by Stachow and Mittelstaedt [33] ,[32], [29], [30]. In these papers they analyse quantum logic using the ideas of game theory. In particular they define logical connectives in terms of sequences of subsequent moves of possible attacks and defenses. They also introduce the concept of `commensurability property' which essentially defines the possibility of quantities being measured at the same time or not. The definition of

**logical connectives**involves both possible attacks and defenses as well as the satisfaction of the commensurability property since logical connective relate propositions which refer to the same time. On the other hand, the definition of**sequential connectives**does not need the introduction of the commensurability properties since sequential connectives refer to propositions defined at different times, and thus can always be evaluated together. The commensurability property introduced by Stachow and Mittelstaedt can be seen as the game theory analogue of the commutation relation between operators in quantum theory. We note that the same type of analysis can be applied as a justification of Isham's choice of the tensor product as temporal connective in the HPO theory..

- ....
^{15} - This is correct since the projectors which appear on the
right hand side of the equation are pair-wise orthogonal, thus the
`or',
, can be replaced by the summation operation
of
projector operators.
.