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This post is a continuation of this post and this post here. If anybody's interested and hasn't read them yet, go there and do so before looking behind the cut.
Having discarded the method of describing a history mathematically by the application of a sequence of projection operators in series to a Hilbert space, like so: Â1Â2...Ân, Isham proposes that a better description of the behaviour of a history proposition is provided by a sequence of tensor products of single time projection operators, Â1⊗Â2⊗...⊗Ân.
According to Isham, this is justified by the fact that each projection operator belongs to a vector space, which is the set of all possible projection operators capable of acting on the Hilbert space, and which is designated here as B(H). A history {Â1,Â2,...,Ân} is part of the direct sum of all the vector spaces which contain each of the projection operators within the history, ⊕t∈{t1,t2,...,tn}B(H).
By way of explanation, a direct sum of two vector spaces is a combination of the two that forms a larger vector space which contains all the dimensions of the original two vectors spaces. For example, the direct sum of a vector space with m dimensions and a vector space with n dimensions would be a vector space with m + n dimensions. Also, it should be noted that the projection operators referred to here are written as matrices, which resemble rectangular tables of numbers. The tensor product of two matrices of similar dimensions is a matrix in which each number is the product of the two numbers in the same position in the original two matrices.
The map of {Â1,Â2,...,Ân}↦tr(Â1(t1)Â2(t2)...Ân(tn)Ê) describes the contribution of the history {Â1,Â2,...,Ân} to the decoherence functional, where the result of the map is one half of the decoherence functional. This map would require multiple steps to be performed, one for each proposition included in the history. It is, however, possible to transform the product of the many vector spaces containing the propositions directly into a tensor product of vector spaces, and from the tensor product transform directly into the vector space of all tr(Â1(t1)Â2(t2)...Ân(tn)Ê). It is suggested by Isham that this result indicates the tensor product to be the natural consequence of the histories approach in quantum theory.
The 'and' logical operation on two histories {Â1,Â2,...,Ân} and {Ê1,Ê2,...,Ên} is thus
{Â1,Â2,...,Ân}∧{Ê1,Ê2,...,Ên} = (Â1⊗Â2⊗...⊗Ân)(Ê1⊗Ê2⊗...⊗Ên) = Â1Ê1⊗Â2Ê2⊗...⊗ÂnÊn
In the case where {Â1,Â2,...,Ân} and {Ê1,Ê2,...,Ên} are disjoint, the proposition '{Â1,Â2,...,Ân} or {Ê1,Ê2,...,Ên} happens' is represented by
{Â1,Â2,...,Ân}∨{Ê1,Ê2,...,Ên} = Â1⊗Â2⊗...⊗Ân + Ê1⊗Ê2⊗...⊗Ên
This result resembles very closely the one obtained using standard quantum theory, as described in the previous post. Similarly, as the Hilbert spaces of all the propositions have now been combined to form the Hilbert space of all histories, it is now possible to say
¬{Â1,Â2,...,Ân} = 1 - Â1⊗Â2⊗...⊗Ân.
Additionally, for the condition in which {Â1,Â2,...,Ân} and {Ê1,Ê2,...,Ên} are not disjoint, the standard result for two such projection operators, Ô and Û which commute is
Ô∨Û = Ô + Û - ÔÛ
Thus,
{Â1,Â2,...,Ân}∨{Ê1,Ê2,...,Ên} = Â1⊗Â2⊗...⊗Ân + Ê1⊗Ê2⊗...⊗Ên - (Â1⊗Â2⊗...⊗Ân)(Ê1⊗Ê2⊗...⊗Ên)
This use of all the logical operators to test the validity of the History Projection Operator formalism does indeed strongly suggest that sequential tensor products of projection operators are indeed accurate mathematical descriptions of histories.
The caveat here, however, is that the projectors here are only of histories in which propositions are made at the same time. That matter requires rather more finesse.
Having discarded the method of describing a history mathematically by the application of a sequence of projection operators in series to a Hilbert space, like so: Â1Â2...Ân, Isham proposes that a better description of the behaviour of a history proposition is provided by a sequence of tensor products of single time projection operators, Â1⊗Â2⊗...⊗Ân.
According to Isham, this is justified by the fact that each projection operator belongs to a vector space, which is the set of all possible projection operators capable of acting on the Hilbert space, and which is designated here as B(H). A history {Â1,Â2,...,Ân} is part of the direct sum of all the vector spaces which contain each of the projection operators within the history, ⊕t∈{t1,t2,...,tn}B(H).
By way of explanation, a direct sum of two vector spaces is a combination of the two that forms a larger vector space which contains all the dimensions of the original two vectors spaces. For example, the direct sum of a vector space with m dimensions and a vector space with n dimensions would be a vector space with m + n dimensions. Also, it should be noted that the projection operators referred to here are written as matrices, which resemble rectangular tables of numbers. The tensor product of two matrices of similar dimensions is a matrix in which each number is the product of the two numbers in the same position in the original two matrices.
The map of {Â1,Â2,...,Ân}↦tr(Â1(t1)Â2(t2)...Ân(tn)Ê) describes the contribution of the history {Â1,Â2,...,Ân} to the decoherence functional, where the result of the map is one half of the decoherence functional. This map would require multiple steps to be performed, one for each proposition included in the history. It is, however, possible to transform the product of the many vector spaces containing the propositions directly into a tensor product of vector spaces, and from the tensor product transform directly into the vector space of all tr(Â1(t1)Â2(t2)...Ân(tn)Ê). It is suggested by Isham that this result indicates the tensor product to be the natural consequence of the histories approach in quantum theory.
The 'and' logical operation on two histories {Â1,Â2,...,Ân} and {Ê1,Ê2,...,Ên} is thus
{Â1,Â2,...,Ân}∧{Ê1,Ê2,...,Ên} = (Â1⊗Â2⊗...⊗Ân)(Ê1⊗Ê2⊗...⊗Ên) = Â1Ê1⊗Â2Ê2⊗...⊗ÂnÊn
In the case where {Â1,Â2,...,Ân} and {Ê1,Ê2,...,Ên} are disjoint, the proposition '{Â1,Â2,...,Ân} or {Ê1,Ê2,...,Ên} happens' is represented by
{Â1,Â2,...,Ân}∨{Ê1,Ê2,...,Ên} = Â1⊗Â2⊗...⊗Ân + Ê1⊗Ê2⊗...⊗Ên
This result resembles very closely the one obtained using standard quantum theory, as described in the previous post. Similarly, as the Hilbert spaces of all the propositions have now been combined to form the Hilbert space of all histories, it is now possible to say
¬{Â1,Â2,...,Ân} = 1 - Â1⊗Â2⊗...⊗Ân.
Additionally, for the condition in which {Â1,Â2,...,Ân} and {Ê1,Ê2,...,Ên} are not disjoint, the standard result for two such projection operators, Ô and Û which commute is
Ô∨Û = Ô + Û - ÔÛ
Thus,
{Â1,Â2,...,Ân}∨{Ê1,Ê2,...,Ên} = Â1⊗Â2⊗...⊗Ân + Ê1⊗Ê2⊗...⊗Ên - (Â1⊗Â2⊗...⊗Ân)(Ê1⊗Ê2⊗...⊗Ên)
This use of all the logical operators to test the validity of the History Projection Operator formalism does indeed strongly suggest that sequential tensor products of projection operators are indeed accurate mathematical descriptions of histories.
The caveat here, however, is that the projectors here are only of histories in which propositions are made at the same time. That matter requires rather more finesse.