Interpretative Aspects of Quantum Mechanics by Matteo Campanella & David Jou & Maria Stella Mongiovì

Author:Matteo Campanella & David Jou & Maria Stella Mongiovì
Language: eng
Format: epub
ISBN: 9783030442071
Publisher: Springer International Publishing

(A.2.1)

with , , and .

The given congruence can be characterized by a mapping which associates to each its class of congruence . Then there is a unique pair in the given representation such that . We have and . If , we have and then . Thus, we get

(A.2.2)

where and with the condition that q is a left factor of .

Putting in (A.2.2) and calling be the projection of on the first factor, we get and then . Hence, the partition is uniquely determined by the congruence and the mapping q with it. Thus, two representations of the same congruence must give rise to the same partition.

We also have for every . If denotes the projection of on the second factor, we get for with . These equations must be satisfied for every representation of the given congruence. Thus, if with and with correspond to two representations of the same congruence, we must have for . Let us represent as and as . Thus, we have . If we take , we see that and belong to the same conjugation class, so that there is such that . Thus, we have , i.e., . Therefore where is the normalizer of . Hence we get with . Then we obtain . We further have . Consequently , i.e., . This means that , that is, that the element of is independent of s for . Thus, there is such that . Consequently, with . Thus . Putting , we have , which entails . Conversely, if , . As q and uniquely determine the congruence, we conclude the following.

The mappings and such thatqis a left factor of and is a left factor of define the same congruence if and only if and there is a mapping factorizable byqon the left such that .

Returning to the initial free G-set we can rewrite (A.2.2) in the form

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