Toward quantum mathematics. I. From quantum set theory to universal quantum mechanics

Abstract

We develop the old idea of von Neumann of a set theory with an internal quantum logic in a modern categorical guise [i.e., taking the objects of the category H of (pre-)Hilbert spaces and linear maps as the sets of the basic level]. We will see that in this way it is possible to clarify the relationship between categorification and quantization and besides this to understand that in some sense a categorificational approach to quantization is a discretized version of the one taken by noncommutative geometry. The tower of higher categorifications will appear as the analog of the von Neumann hierarchy of classical set theory (where by classical set theory, we will understand the usual Zermelo–Fraenkel system). Finally, we make a suggestion of how to understand all the different categorifications as different realizations of one and the same abstract structure by viewing quantum mechanics as universal in the sense of category theory. This gives the possibility to view extended topological quantum field theories purely as involving an abstract notion of quantum mechanics plus representation theory without the need to enlarge the class of kinematic structures of quantum systems on each step of categorification. In a future part of the work we will apply the language developed here to deal especially with the question of a categorification of the manifold notion.

Reference frame fields based on quantum theory representations of real and complex numbers

“…Other work which emphasizes the close relationship between physics and mathematics is concerned with quantum theory representations of mathematical systems. This work includes papers on quantum set theory [10,11,12], quantum theory representations of real numbers [13,14,15,16,17,18,19], and the use of category theory in physics [20,21].…”

“…This follows from the observations that for each k V k j =Ṽ j+1 (11) and that the implementation of eachṼ j is efficient. Also implementation of the various arithmetic operations by use of theṼ j is efficient.…”

Η Λογική Της Μετάβασης Από Την Κλασική Στην Κβαντική Φυσική Υπό Το Πρίσμα Σύγχρονων Μαθηματικών Εξελίξεων

“…Αυτές οι διαφορές στην γλώσσα της θεωρίας των κατηγοριών, ισοδυναμούν με την παρουσία υπερθέσεων που παίζουν τόσο καίριο ρόλο και εκφράζουν την κβαντική μη διαχωρισιμότητα [37,41]. Μια άλλη συνέπεια είναι ότι σε μια (καθαρή) συνεζευγμένη κατάσταση ενός σύνθετου κβαντικού συστήματος δεν υφίστανται (καθαρές) καταστάσεις των μερών του [42].…”

“…Μια προσεταιριστική άλγεβρα Α είναι ∧ 2-βαθμωτή-μεταθετική (graded commutative), ή απλώς υπερμεταθετική (supercommutative), 41…”

“…Μολονότι κορυφαίοι φυσικοί, όπως ο R. Haag, επιμένουν στην αυτοτελή θεμελίωση της αξιωματικής διατύπωσης. Βλέπε, π.χ.,[92] 41. Το πρόθεμα «υπέρ» σημαίνει ότι έχουμε στοιχεία που μετατίθενται και στοιχεία που αντιμετατίθενται.…”