We derive some Quantum Central Limit Theorems for expectation values of macroscopically coarse-gr... more We derive some Quantum Central Limit Theorems for expectation values of macroscopically coarse-grained observables, which are functions of coarse-grained hermitean operators. Thanks to the hermicity constraints, we obtain positive-definite distribution for the expectation values of observables. These probability distributions open some pathway for an emergence of classical behaviours in the limit of infinitely large number of identical and non-interacting quantum constituents. This is in contradistinction to other mechanisms of classicality emergence due to environmental decoherence and consistent histories. The probability distributions so derived also enable us to evaluate the nontrivial time-dependence of certain differential entropies.
The path-integral functional of chiral gauge theories with background gauge potentials are derive... more The path-integral functional of chiral gauge theories with background gauge potentials are derived in the holomorphic representation. Justification is provided, from first quantum mechanical principles, for the appearance of a functional phase factor of the gauge fields in order to maintain the gauge invariance. This term is shown to originate either from the Berry phase of the first-quantized hamiltonians or from the normal ordering of the second-quantized hamiltonian with respect to the Dirac in-vacuum. The quantization of the chiral Schwinger model is taken as an example.
A method is presented to tackle the sign problem in the simulations of systems having inde nite o... more A method is presented to tackle the sign problem in the simulations of systems having inde nite or complex-valued measures. In general, this new approach is shown to yield statistical errors smaller than the crude Monte Carlo using absolute values of the original measures. Exactly solvable, one-dimensional Ising models with complex temperature and complex activity illustrate the considerable improvements and the workability of the new method even when the crude one fails.
Abstract We propose a new class of permanent magnetic lattices on an atom chip for producing 1D a... more Abstract We propose a new class of permanent magnetic lattices on an atom chip for producing 1D and 2D periodic arrays of Ioffe-Pritchard magnetic micro-traps with variable barrier height for trapping and manipulating ultracold atoms and Bose-Einstein condensates. The ...
International Journal of Modern Physics, Jan 10, 1992
A formulation is proposed of Abelian chiral gauge theory which is invariant with respect to a gau... more A formulation is proposed of Abelian chiral gauge theory which is invariant with respect to a gauge symmetry and admits both fermion and vector-boson mass terms, without invoking the Higgs mechanism. The issues of unitarity and renormalizability are discussed, and a lattice chiral regularization free from the problem of fermion-species doubling is constructed and compared with others.
The chiral Schwinger model is formulated in the wilson-fermion formulation on the lattice and the... more The chiral Schwinger model is formulated in the wilson-fermion formulation on the lattice and then simulated by the complex langevin algorithm. The simulation is done both without and with gauge xing to the Lorentz gauge for the compact gauge links. Some preliminary results are presented which indicate that the complex langevin is well behaving with the complex chiral fermion determinant.
Random-lattice fermions have been shown to be free of the doubling problem if there are no intera... more Random-lattice fermions have been shown to be free of the doubling problem if there are no interactions or interactions of a nongauge nature. However, gauge interactions impose stringent constraints as expressed by the Ward-Takahashi identities which could revive the free-field suppressed doubler modes in loop diagrams. After introducing a formulation for fermions on a new kind of random lattice, we compare random, naive and Wilson fermions in two dimensional Abelian background gauge theory. We show that the doublers are revived for random lattices in the continuum limit, while demonstrating that gauge invariance plays the critical role in this revival. Some implications of the persistent doubling phenomenon on random lattices are also discussed.
We employ quantum mechanical principles in the computability exploration of the class of classica... more We employ quantum mechanical principles in the computability exploration of the class of classically noncomputable Hilbert's tenth problem which is equivalent to the Turing halting problem in Computer Science. The Quantum Adiabatic Theorem enables us to establish a connection between the solution for this class of problems and the asymptotic behaviour of solutions of a particular type of time-dependent Schrödinger equations. We then present some preliminary numerical simulation results for the quantum adiabatic processes corresponding to various Diophantine equations.
... fluctuations were correlated by deriving the two fields from an original phase-fluctuating fi... more ... fluctuations were correlated by deriving the two fields from an original phase-fluctuating field using acousto ... This restoration of the coherence hole for correlated fluctuating laser fields described by the Wiener-Levy phase-diffusion model was predicted by Dalton and Knight [9 ...
Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, Jun 1, 2019
A new class of time-energy uncertainty relations is directly derived from the Schrödinger equatio... more A new class of time-energy uncertainty relations is directly derived from the Schrödinger equations for time-dependent Hamiltonians. Only the initial states and the Hamiltonians, but neither the instantaneous eigenstates nor the full time-dependent wave functions, which would demand a full solution for a time-dependent Hamiltonian, are required for our time-energy relations. Explicit results are then presented for particular subcases of interest for timeindependent Hamiltonians and also for time-varying Hamiltonians employed in adiabatic quantum computation. Some estimates of the lower bounds on computational time are given for general adiabatic quantum algorithms, with Grover's search as an illustration. We particularly emphasize the role of required energy resources, besides the space and time complexity, for the physical process of (quantum) computation, in general.
Despite the recursive non-computability of Hilbert's tenth problem, we outline and argue for a qu... more Despite the recursive non-computability of Hilbert's tenth problem, we outline and argue for a quantum algorithm that is based on the Quantum Adiabatic Theorem. It is explained how this algorithm can solve Hilbert's tenth problem. The algorithm is then considered in the context of several "no-go" arguments against such hypercomputation. Logical arguments are usually based on Cantor's diagonal technique used for proving non-computability of the Turing halting problem, which is related to Hilbert's tenth problem. Physical arguments are related to the limited computability of a class of quantum computation based on qubits and dimensionally ÿnite quantum logical gates.
International Journal of Modern Physics C, Apr 1, 1994
To tackle the sign problem in the simulations of systems having inde nite or complex-valued measu... more To tackle the sign problem in the simulations of systems having inde nite or complex-valued measures, we propose a new approach which yields statistical errors smaller than the crude Monte Carlo using absolute values of the original measures. The 1D complex-coupling Ising model is employed as an illustration.
The conditions for the experimental observation of new types of eigenmodes in a typical laborator... more The conditions for the experimental observation of new types of eigenmodes in a typical laboratory magnetized plasma with two ion species are discussed. It is shown that the most favourable condition occurs during the current carrying phase of the discharge, with an appropriately chosen mixture of ions.
A non-polynomial quantization for chiral gauge interactions is motivated. It is shown for Abelian... more A non-polynomial quantization for chiral gauge interactions is motivated. It is shown for Abelian group in four dimensions that the quantization, manifestly local in the Lorentz gauge, is consistent: simultaneously satisfying the requirements of gauge invariance, (perturbative) renormalizability, and unitarity. Chiral photons can be massive in the framework without invoking the Higgs mechanism. Non-Abelian generalization is then speculated.
We derive some Quantum Central Limit Theorems for expectation values of macroscopically coarse-gr... more We derive some Quantum Central Limit Theorems for expectation values of macroscopically coarse-grained observables, which are functions of coarse-grained hermitean operators. Thanks to the hermicity constraints, we obtain positive-definite distribution for the expectation values of observables. These probability distributions open some pathway for an emergence of classical behaviours in the limit of infinitely large number of identical and non-interacting quantum constituents. This is in contradistinction to other mechanisms of classicality emergence due to environmental decoherence and consistent histories. The probability distributions so derived also enable us to evaluate the nontrivial time-dependence of certain differential entropies.
The path-integral functional of chiral gauge theories with background gauge potentials are derive... more The path-integral functional of chiral gauge theories with background gauge potentials are derived in the holomorphic representation. Justification is provided, from first quantum mechanical principles, for the appearance of a functional phase factor of the gauge fields in order to maintain the gauge invariance. This term is shown to originate either from the Berry phase of the first-quantized hamiltonians or from the normal ordering of the second-quantized hamiltonian with respect to the Dirac in-vacuum. The quantization of the chiral Schwinger model is taken as an example.
A method is presented to tackle the sign problem in the simulations of systems having inde nite o... more A method is presented to tackle the sign problem in the simulations of systems having inde nite or complex-valued measures. In general, this new approach is shown to yield statistical errors smaller than the crude Monte Carlo using absolute values of the original measures. Exactly solvable, one-dimensional Ising models with complex temperature and complex activity illustrate the considerable improvements and the workability of the new method even when the crude one fails.
Abstract We propose a new class of permanent magnetic lattices on an atom chip for producing 1D a... more Abstract We propose a new class of permanent magnetic lattices on an atom chip for producing 1D and 2D periodic arrays of Ioffe-Pritchard magnetic micro-traps with variable barrier height for trapping and manipulating ultracold atoms and Bose-Einstein condensates. The ...
International Journal of Modern Physics, Jan 10, 1992
A formulation is proposed of Abelian chiral gauge theory which is invariant with respect to a gau... more A formulation is proposed of Abelian chiral gauge theory which is invariant with respect to a gauge symmetry and admits both fermion and vector-boson mass terms, without invoking the Higgs mechanism. The issues of unitarity and renormalizability are discussed, and a lattice chiral regularization free from the problem of fermion-species doubling is constructed and compared with others.
The chiral Schwinger model is formulated in the wilson-fermion formulation on the lattice and the... more The chiral Schwinger model is formulated in the wilson-fermion formulation on the lattice and then simulated by the complex langevin algorithm. The simulation is done both without and with gauge xing to the Lorentz gauge for the compact gauge links. Some preliminary results are presented which indicate that the complex langevin is well behaving with the complex chiral fermion determinant.
Random-lattice fermions have been shown to be free of the doubling problem if there are no intera... more Random-lattice fermions have been shown to be free of the doubling problem if there are no interactions or interactions of a nongauge nature. However, gauge interactions impose stringent constraints as expressed by the Ward-Takahashi identities which could revive the free-field suppressed doubler modes in loop diagrams. After introducing a formulation for fermions on a new kind of random lattice, we compare random, naive and Wilson fermions in two dimensional Abelian background gauge theory. We show that the doublers are revived for random lattices in the continuum limit, while demonstrating that gauge invariance plays the critical role in this revival. Some implications of the persistent doubling phenomenon on random lattices are also discussed.
We employ quantum mechanical principles in the computability exploration of the class of classica... more We employ quantum mechanical principles in the computability exploration of the class of classically noncomputable Hilbert's tenth problem which is equivalent to the Turing halting problem in Computer Science. The Quantum Adiabatic Theorem enables us to establish a connection between the solution for this class of problems and the asymptotic behaviour of solutions of a particular type of time-dependent Schrödinger equations. We then present some preliminary numerical simulation results for the quantum adiabatic processes corresponding to various Diophantine equations.
... fluctuations were correlated by deriving the two fields from an original phase-fluctuating fi... more ... fluctuations were correlated by deriving the two fields from an original phase-fluctuating field using acousto ... This restoration of the coherence hole for correlated fluctuating laser fields described by the Wiener-Levy phase-diffusion model was predicted by Dalton and Knight [9 ...
Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, Jun 1, 2019
A new class of time-energy uncertainty relations is directly derived from the Schrödinger equatio... more A new class of time-energy uncertainty relations is directly derived from the Schrödinger equations for time-dependent Hamiltonians. Only the initial states and the Hamiltonians, but neither the instantaneous eigenstates nor the full time-dependent wave functions, which would demand a full solution for a time-dependent Hamiltonian, are required for our time-energy relations. Explicit results are then presented for particular subcases of interest for timeindependent Hamiltonians and also for time-varying Hamiltonians employed in adiabatic quantum computation. Some estimates of the lower bounds on computational time are given for general adiabatic quantum algorithms, with Grover's search as an illustration. We particularly emphasize the role of required energy resources, besides the space and time complexity, for the physical process of (quantum) computation, in general.
Despite the recursive non-computability of Hilbert's tenth problem, we outline and argue for a qu... more Despite the recursive non-computability of Hilbert's tenth problem, we outline and argue for a quantum algorithm that is based on the Quantum Adiabatic Theorem. It is explained how this algorithm can solve Hilbert's tenth problem. The algorithm is then considered in the context of several "no-go" arguments against such hypercomputation. Logical arguments are usually based on Cantor's diagonal technique used for proving non-computability of the Turing halting problem, which is related to Hilbert's tenth problem. Physical arguments are related to the limited computability of a class of quantum computation based on qubits and dimensionally ÿnite quantum logical gates.
International Journal of Modern Physics C, Apr 1, 1994
To tackle the sign problem in the simulations of systems having inde nite or complex-valued measu... more To tackle the sign problem in the simulations of systems having inde nite or complex-valued measures, we propose a new approach which yields statistical errors smaller than the crude Monte Carlo using absolute values of the original measures. The 1D complex-coupling Ising model is employed as an illustration.
The conditions for the experimental observation of new types of eigenmodes in a typical laborator... more The conditions for the experimental observation of new types of eigenmodes in a typical laboratory magnetized plasma with two ion species are discussed. It is shown that the most favourable condition occurs during the current carrying phase of the discharge, with an appropriately chosen mixture of ions.
A non-polynomial quantization for chiral gauge interactions is motivated. It is shown for Abelian... more A non-polynomial quantization for chiral gauge interactions is motivated. It is shown for Abelian group in four dimensions that the quantization, manifestly local in the Lorentz gauge, is consistent: simultaneously satisfying the requirements of gauge invariance, (perturbative) renormalizability, and unitarity. Chiral photons can be massive in the framework without invoking the Higgs mechanism. Non-Abelian generalization is then speculated.
We derive some Quantum Central Limit Theorems for expectation values of macroscopically coarse-gr... more We derive some Quantum Central Limit Theorems for expectation values of macroscopically coarse-grained observables, which are functions of coarse-grained hermitean operators consisting of non-commuting variables. Thanks to the hermicity constraints, we obtain positive-definite distribution for the expectation values of observables. These probability distributions open some pathway for an emergence of classical behaviours in the limit of infinitely large number of identical and non-interacting quantum constituents. This is in contradistinction to other mechanisms of classicality emergence due to environmental decoherence and consistent histories. The probability distributions so derived also enable us to evaluate the nontrivial time-dependence of certain differential entropies.
The Principle of Unattainability rules out the attainment of absolute zero temperature by any phy... more The Principle of Unattainability rules out the attainment of absolute zero temperature by any physical means, no matter how idealised they could be. Nevertheless, we clarify that the Third Law of Thermodynamics, as defined by Nernst's heat theorem statement, is distinct from the Principle of Unattainability in the sense that the Third Law is mathematically equivalent only to the unattainability of absolute zero temperature by quasi-static adiabatic processes. This thus leaves open the possibility of attainability of absolute zero, without violating the Third Law, by non-adiabatic means. Such a means may be provided in principle and in particular by projective measurements in quantum mechanics. This connection also establishes some intimate relationship between the postulate of projective measurement and the Principle of Unattainability. * tien.d.kieu@gmail.com
A new class of time-energy uncertainty relations is directly derived from the Schrödinger equatio... more A new class of time-energy uncertainty relations is directly derived from the Schrödinger equations for time-dependent Hamiltonians. Explicit results are then presented for particular subcases of interest for time-independent Hamiltonians and also for time-varying Hamiltonians employed in quantum adiabatic computation. Similar to the position-momentum uncertainty relation (PMUR), the time-energy uncertainty relation (TEUR) is a property of the conjugacy of the Fourier transform variables time and frequency and of the quantum relationship between energy and frequency. But unlike the PMUR, the TEUR cannot be derived from any commutator relation, due to the lack of a well-defined time operator in quantum mechanics. As a consequence, the interpretation of the TEUR is not as simple as that of the PMUR and has lead to serious misinterpretations [10]. The uncertainty relation between energy and time in nonrelativistic quantum mechanics was derived [7, 8] as the time it takes for a state with given energy spread to decay as measured through the rate of change of the expectation value of some observable. There are also other time energy relations expressing the least time it takes for the system under investigation to evolve into an orthogonal state in terms of the energy and energy spread of the system [2, 3, 9, 11]. In this paper we derive different sets of TEURs directly from the Schrödinger equation for time-dependent Hamil-tonians in the general case. Our employed method of derivation as well as the results are new and different to those in the existing literature. As a special case for time-independent Hamiltonians, and as a necessary but not sufficient condition, is a lower bound on the time ∆t ∀ it takes for an initial state to evolve into any arbitrary state allowed by the Hamiltonian dynamics: 2¯ h ≤ ∆t ∀ × ∆E 0 2 + (E 0) 2 , (1) where E 0 and ∆E 0 are, respectively the energy expectation value and the energy spread of the initial state. Also, as a necessary but not sufficient condition is a relation between the energy and the lower bound on the time ∆t ⊥ it takes for an initial state to evolve into an orthogonal state allowed by the Hamiltonian dynamics: ¯ h √ 2 ≤ ∆t ⊥ × ∆E 0. (2) The necessary conditions above can also be expressed differently but equivalently as that the system cannot fully explore the whole Hilbert space (that is, cannot reach certain dynamically allowable state) or evolve into an orthogonal state from the initial state if the evolution time is less than, respectively, the characteristic times T ∀ or T ⊥ : T ∀ ≡ 2¯ h/ ∆E 0 2 + (E 0) 2 , T ⊥ ≡ ¯ h √ 2/∆E 0. (3) In addition, we also obtain some results for the quantum adiabatic computation QAC [4, 5] with time-varying Hamiltonians. QAC starts with the readily constructible ground state |g I of an initial Hamiltonian H I which is then adiabatically extrapolated to the final Hamiltonian H P whose ground state |g encodes the desirable solution of the problem and could be then obtained with reasonably high probability. The interpolation between H I and H P is facilitated by a time-dependent Hamiltonian in the time interval 0 ≤ t ≤ T , H(t) = f (t/T)H I + g(t/T)H P , (4) either in a temporally linear manner (that is, f (t/T) = (1 − t/T) and g(t/T) = t/T); or otherwise with f (0) = 1 = g(1) and f (1) = 0 = g(0). We also assume that both f and g are continuous, and g is semi-positive for all t ∈ [0, T ]. Such a time evolution is captured by the following Schrödinger equation: i∂ t |ψ(t) = H(t) |ψ(t), (5) |ψ(0) = |g I .
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