Browsing by Author "Lewis, Zachary"
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- Bell's Inequalities, Superquantum Correlations, and String TheoryChang, Lay Nam; Lewis, Zachary; Minic, Djordje; Takeuchi, Tatsu; Tze, Chia-Hsiung (Hindawi, 2011-01-01)We offer an interpretation of super-quantum correlations in terms of a “doubly” quantum theory. We argue that string theory, viewed as a quantum theory with two deformation parameters, the string tension a′ and the string coupling constant gs, is such a super-quantum theory, one that transgresses the usual quantum violations of Bell’s inequalities. We also discuss the ħ ⟶ ∞ limit of quantum mechanics in this context. As a super-quantum theory, string theory should display distinct experimentally observable super-correlations of entangled stringy states.
- Bell's Inequalities, Superquantum Correlations, and String TheoryChang, Lay Nam; Lewis, Zachary; Minic, Djordje; Takeuchi, Tatsu; Tze, Chia-Hsiung (Hindawi Publishing Corporation, 2011)We offer an interpretation of superquantum correlations in terms of a “doubly” quantum theory.We argue that string theory, viewed as a quantum theory with two deformation parameters, the string tension α', and the string coupling constant gs, is such a superquantum theory that transgresses the usual quantum violations of Bell's inequalities. We also discuss the ℏ→∞ limit of quantum mechanics in this context. As a superquantum theory, string theory should display distinct experimentally observable supercorrelations of entangled stringy states.
- Biorthogonal quantum mechanics: super-quantum correlations and expectation values without definite probabilitiesChang, Lay Nam; Lewis, Zachary; Minic, Djordje; Takeuchi, Tatsu (IOP, 2013-12-06)We propose mutant versions of quantum mechanics constructed on vector spaces over the finite Galois fields GF(3) and GF(9). The mutation we consider here is distinct from what we proposed in previous papers on Galois field quantum mechanics. In this new mutation, the canonical expression for expectation values is retained instead of that for probabilities. In fact, probabilities are indeterminate. Furthermore, it is shown that the mutant quantum mechanics over the finite field GF(9) exhibits super-quantum correlations (i.e. the Bell-Clauser-Horne-Shimony-Holt bound is 4). We comment on the fundamental physical importance of these results in the context of quantum gravity.
- Galois Field Quantum MechanicsChang, Lay Nam; Lewis, Zachary; Minic, Djordje; Takeuchi, Tatsu (World Scientific, 2013-04-20)We construct a discrete quantum mechanics using a vector space over the Galois field GF(q). We find that the correlations in our model do not violate the Clauser-Horne-Shimony-Holt (CHSH) version of Bell’s inequality, despite the fact that the predictions of this discrete quantum mechanics cannot be reproduced with any hidden variable theory.
- Is Quantum Gravity a Super-Quantum Theory?Chang, Lay Nam; Lewis, Zachary; Minic, Djordje; Takeuchi, Tatsu (World Scientific, 2013-10-01)We argue that quantum gravity should be a super-quantum theory, that is, a theory whose non-local correlations are stronger than those of canonical quantum theory. As a super-quantum theory, quantum gravity should display distinct experimentally observable super-correlations of entangled stringy states.
- On the Minimal Length Uncertainty Relation and the Foundations of String TheoryChang, Lay Nam; Lewis, Zachary; Minic, Djordje; Takeuchi, Tatsu (Hindawi Publishing Corporation, 2011)We review our work on the minimal length uncertainty relation as suggested by perturbative string theory. We discuss simple phenomenological implications of the minimal length uncertainty relation and then argue that the combination of the principles of quantum theory and general relativity allow for a dynamical energy-momentum space. We discuss the implication of this for the problem of vacuum energy and the foundations of nonperturbative string theory.
- On the Minimal Length Uncertainty Relation and the Foundations of String TheoryChang, Lay Nam; Lewis, Zachary; Minic, Djordje; Takeuchi, Tatsu (Hindawi, 2011-01-01)We review our work on the minimal length uncertainty relation as suggested by perturbative string theory. We discuss simple phenomenological implications of the minimal length uncertainty relation and then argue that the combination of the principles of quantum theory and general relativity allow for a dynamical energy-momentum space. We discuss the implication of this for the problem of vacuum energy and the foundations of non-perturbative string theory.
- Position and momentum uncertainties of a particle in a V-shaped potential under the minimal length uncertainty relationLewis, Zachary; Roman, Ahmed; Takeuchi, Tatsu (World Scientific, 2015-12-20)
- Position and momentum uncertainties of the normal and inverted harmonic oscillators under the minimal length uncertainty relationLewis, Zachary; Takeuchi, Tatsu (American Physical Society, 2011-11-18)We analyze the position and momentum uncertainties of the energy eigenstates of the harmonic oscillator in the context of a deformed quantum mechanics, namely, that in which the commutator between the position and momentum operators is given by [(x) over cap, (p) over cap] = i (h) over bar (1 + beta(p) over cap (2)). This deformed commutation relation leads to the minimal length uncertainty relation Delta x >= ((h) over bar /2)(1/Delta p + beta Delta p), which implies that Delta x similar to 1/Delta p at small Delta p while Delta x similar to Delta p at large Delta p. We find that the uncertainties of the energy eigenstates of the normal harmonic oscillator (m > 0), derived in L. N. Chang, D. Minic, N. Okamura, and T. Takeuchi, Phys. Rev. D 65, 125027 ( 2002), only populate the Delta x similar to 1/Delta p branch. The other branch, Delta x similar to Delta p, is found to be populated by the energy eigenstates of the "inverted" harmonic oscillator (m < 0). The Hilbert space in the inverted case admits an infinite ladder of positive energy eigenstates provided that Delta x(min) = <(h)over bar>root beta > root 2[(h) over bar (2)/k vertical bar m vertical bar](1/4). Correspondence with the classical limit is also discussed.
- Position and momentum uncertainties of the normal and inverted harmonic oscillators under the minimal length uncertainty relationLewis, Zachary; Takeuchi, Tatsu (American Physical Society, 2011-11-18)
- Quantum F-un: the q=1 limit of Galois field quantum mechanics, projective geometry and the field with one elementChang, Lay Nam; Lewis, Zachary; Minic, Djordje; Takeuchi, Tatsu (IOP, 2014-10-10)We argue that the q = 1 limit of Galois Field Quantum Mechanics, which was constructed on a vector space over the Galois Field Fq = GF(q), corresponds to its ‘classical limit,’ where superposition of states is disallowed. The limit preserves the projective geometry nature of the state space, and can be understood as being constructed on an appropriately defined analogue of a ‘vector’ space over the ‘field with one element’ F1.
- Quantum Systems Based Upon Galois Fields - From Sub-Quantum to Super-Quantum CorrelationsChang, Lay Nam; Lewis, Zachary; Minic, Djordje; Takeuchi, Tatsu (World Scientific, 2014-02-10)In this talk we describe our recent work on discrete quantum theory based on Galois fields. In particular, we discuss how discrete quantum theory sheds new light on the foundations of quantum theory and we review an explicit model of super-quantum correlations we have constructed in this context. We also discuss the larger questions of the origins and foundations of quantum theory, as well as the relevance of super-quantum theory for the quantum theory of gravity.
- Some Mutant Forms of Quantum MechanicsTakeuchi, Tatsu; Chang, Lay Nam; Lewis, Zachary; Minic, Djordje (American Institute of Physics, 2012-01-01)We construct a ‘mutant’ form of quantum mechanics on a vector space over the finite Galois field GF(q). We find that the correlations in our model do not violate the Clauser-Horne-Shimony-Holt (CHSH) version of Bell’s inequality, despite the fact that the predictions of this discretized quantum mechanics cannot be reproduced with any hidden variable theory. An alternative ‘mutation’ is also suggested.
- Spin and rotations in Galois field quantum mechanicsChang, Lay Nam; Lewis, Zachary; Minic, Djordje; Takeuchi, Tatsu (IOP, 2013-02-15)We discuss the properties of Galois Field Quantum Mechanics constructed on a vector space over the finite Galois field GF(q). In particular, we look at 2-level systems analogous to spin, and discuss how SO(3) rotations could be embodied in such a system. We also consider two-particle `spin' correlations and show that the Clauser-Horne-Shimony-Holt (CHSH) inequality is nonetheless not violated in this model.
- A Study of Modifications to Quantum MechanicsLewis, Zachary (Virginia Tech, 2013-03-05)In this work, the consequences of several modifications to quantum mechanics are examined. These modifications, motivated by string theory, fall into two categories: ones in which the canonical commutation relations between position and momentum are deformed and ones in which the space of states used are vector spaces over Galois fields instead of complex Hilbert spaces. The particular deformation of the canonical commutation relations used leads to a minimum value of the uncertainty in position which is interpreted as a minimum length scale. Both harmonic and anharmonic oscillators are studied in this framework with normalizable, positive energy eigenstates found in both cases. The quantum uncertainty relations and classical counterparts to these states are discussed. Creating modified quantum theories by replacing the Hilbert spaces of canonical quantum mechanics with vector spaces defined over several finite, Galois fields is accomplished. Correlation functions are calculated in these theories and the maximum values are shown to not behave as would be expected by the standard, Bell-like, bounding inequality theorems. The interpretations and implications of these theories are discussed.