Some mutant forms of quantum mechanics

Takeuchi, Tatsu; Chang, Lay Nam; Lewis, Zachary; Minić, Djordje

doi:10.1063/1.4773173

Cited by 5 publications

(8 citation statements)

References 12 publications

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“…In a Hilbert space, the inner product is used to define the brackets ⟨v i |v⟩ and the norm ∥v∥ = √ ⟨v|v⟩ but there are no inner products in vector spaces over finite fields. The different attempts to develop a toy model of QM over a finite field ([18], [21], [12]) such as Z 2 differ from this model in how they address this problem. The treatment of the Dirac brackets and norm defined here is distinguished by the fact that the resulting probability calculus in QM/Sets is (a non-commutative version of) classical finite probability theory (instead of just a modal calculus with values 0 and 1).…”

Section: The Brackets and Normmentioning

confidence: 99%

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Quantum mechanics over sets: a pedagogical model with non-commutative finite probability theory as its quantum probability calculus

Ellerman

2016

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This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of C replaced by Z2. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts all required the brackets to take values in Z2. But the usual QM brackets ⟨ψ|φ⟩ give the "overlap" between states ψ and φ, so for subsets S, T ⊆ U , the natural definition is ⟨S|T ⟩ = |S ∩ T | (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell's Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over C and QM/Sets over Z2.

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Section: The Brackets and Normmentioning

confidence: 99%

“…There have been at least three previous attempts at developing a version of QM where the base field of C is replaced by Z 2 ([18], [12], and [21]). Since there are no inner products in vector spaces over a finite field, the "trick" is how to define the brackets, the norm, and then the probability algorithm.…”

Section: Introductionmentioning

confidence: 99%

Quantum mechanics over sets: a pedagogical model with non-commutative finite probability theory as its quantum probability calculus

Ellerman

2016

Synthese

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“…There have been at least three previous attempts at developing a version of QM where the base field of C is replaced by Z 2 ([28], [17], and [31]). Since there are no inner products in vector spaces over a finite field, the "trick" is how to define the brackets, the norm, and then the probability algorithm.…”

Section: Dsds Cscos and Measurementmentioning

confidence: 99%

“…In a Hilbert space, the inner product is used to define the brackets v i |v and the norm v = v|v but there are no inner products in vector spaces over finite fields. The different attempts to develop a toy model of QM over a finite field ( [28], [31], [17]) such as Z 2 differ from this model in how they address this problem. The treatment of the Dirac brackets and norm defined here is distinguished by the fact that the resulting probability calculus in QM/Sets is (a non-commutative version of) classical finite probability theory (instead of just a modal calculus with values 0 and 1).…”

Section: The Brackets and The Normmentioning

confidence: 99%

The Quantum Logic of Direct-Sum Decompositions

Ellerman

2016

SSRN Journal

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Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space-which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of partitions to the quantum logic of direct-sum decompositions (i.e., the vector space version of a set partition) of a general vector space-which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the logic to express measurement by any self-adjoint operators rather than just the projection operators associated with subspaces. In this introductory paper, the focus is on the quantum logic of direct-sum decompositions of a finite-dimensional vector space (including such a Hilbert space). The primary special case examined is finite vector spaces over Z2 where the pedagogical model of quantum mechanics over sets (QM/Sets) is formulated. In the Appendix, the combinatorics of direct-sum decompositions of finite vector spaces over GF (q) is analyzed with computations for the case of QM/Sets where q = 2. 1 2 (3 2 −1−2 2 ) = 7×3 3 2 1 2 (4) = 7 × 4 = 28. DSDs in DSD Z 3 2 = all binary DSDs for Z 3 2 .

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“…Finally, let me mention two papers and an ongoing AdS/CFT related project on the physics of the Riemann zeros [32], conducted in collaboration with Yang-Hui He (Oxford/London) and Jejjala (an off-spring of our interest in analytic QCD and many-body physics), and a paper in preparation on the subject of emergent "quantum" theory in the context of biophysical systems, with Vijay Balasubramanian (Penn) and Sinisa Pajevic (NIH). [33][34][35][36][37][38][39][40], 3 are currently undergoing review [41][42][43], and 3 have appeared in conference proceedings [44][45][46]. In addition, 6 more papers are in the final stages of completion in which DOE support will be acknowledged [47][48][49][50][51][52].…”

Section: Applications Of String Theory and Qft In Many-body Physicsmentioning

confidence: 99%