A time-symmetric of quantum mechanics

Update: A time-independent $n\times n$ PT-symmetric (and symmetric) Hamiltonian is diagonalizable since it has all distinct real eigenvalues and the resulting diagonal matrix is a real symmetric matrix. The diagonalization results an isometry so there shouldn't be any issue with unitarity and unfortunately this very elementary mathematical fact somehow did not draw the authors' attention. However, PT-symmetric quantum mechanics is not out of trouble. For time-dependent PT-symmetric (and symmetric) Hamiltonians (even $2\times 2$ ones) the authors observed that there is a violation of unitarity. Moreover, the first named author showed in his recent article arXiv:1312.7738 that PT-symmetric quantum mechanics is indeed a certain kind of Hermitian quantum mechanics and that in order for time-evolution to be unitary with respect to $J$ -inner product (one that gives rise to a Hilbert space structure on the space of state functions), the potential energy operator $V(x)$ must be real. This means that those complex PT-symmetric Hamiltonians that have been studied by physicists are unfortunately unphysical. The first named author discussed in a subsequent article arXiv:1401.5149 that while finite-state PT-symmetric quantum mechanics with time-independent Hamiltonians is not physically any different from Hermitian quantum mechanics, PT-symmetric quantum mechanics exhibits a distinctive symmetry from that of Hermitian quantum mechanics.Update: A time-independent $n\times n$ PT-symmetric (and symmetric) Hamiltonian is diagonalizable since it has all distinct real eigenvalues and the resulting diagonal matrix is a real symmetric matrix. The diagonalization results an isometry so there shouldn't be any issue with unitarity and unfortunately this very elementary mathematical fact somehow did not draw the authors' attention. However, PT-symmetric quantum mechanics is not out of trouble. For time-dependent PT-symmetric (and symmetric) Hamiltonians (even $2\times 2$ ones) the authors observed that there is a violation of unitarity. Moreover, the first named author showed in his recent article arXiv:1312.7738 that PT-symmetric quantum mechanics is indeed a certain kind of Hermitian quantum mechanics and that in order for time-evolution to be unitary with respect to $J$-inner product (one that gives rise to a Hilbert space structure on the space of state functions), the potential energy operator $V(x)$ must be real. This means that those complex PT-symmetric Hamiltonians that have been studied by physicists are unfortunately unphysical. The first named author discussed in a subsequent article arXiv:1401.5149 that while finite-state PT-symmetric quantum mechanics with time-independent Hamiltonians is not physically any different from Hermitian quantum mechanics, PT-symmetric quantum mechanics exhibits a distinctive symmetry from that of Hermitian quantum mechanics.

arXiv:1303.6339Mar 25, 20130 citations

Time evolution and adiabatic approximation in $PT$ -symmetric quantum mechanicsTime evolution and adiabatic approximation in $PT$-symmetric quantum mechanics

Zhihua Guo, Cao, Huaixin

In this paper, we discuss time evolution and adiabatic approximation in $PT$ -symmetric quantum mechanics. we give the time evolving equation for a class of $PT$ -symmetric Hamiltonians and some conditions of the adiabatic approximation for the class of $PT$ -symmetric Hamiltonians.In this paper, we discuss time evolution and adiabatic approximation in $PT$-symmetric quantum mechanics. we give the time evolving equation for a class of $PT$-symmetric Hamiltonians and some conditions of the adiabatic approximation for the class of $PT$-symmetric Hamiltonians.

arXiv:1212.4615Dec 19, 20121 citation

Time Symmetry and Interpretation of Quantum Mechanics

O. Costa de Beauregard

Boston studies in the philosophy of scienceJan 1, 198456 citations

$PT$ -symmetric quantum mechanics is a Hermitian quantum mechanics$PT$-symmetric quantum mechanics is a Hermitian quantum mechanics

Sungwook Lee

The author discusses a different kind of Hermitian quantum mechanics, called $J$ -Hermitian quantum mechanics. He shows that $PT$ -symmetric quantum mechanics is indeed $J$ -Hermitian quantum mechanics, and that time evolution (in the Krein space of states) is unitary if and only if Hamiltonian is $J$ -Hermitian (or equivalently $PT$ -symmetric). An issue with unitarity comes up when time evolution is considered in the Hilbert space of states rather than in the Krein space of states. The author offers possible scenarios with this issue.The author discusses a different kind of Hermitian quantum mechanics, called $J$-Hermitian quantum mechanics. He shows that $PT$-symmetric quantum mechanics is indeed $J$-Hermitian quantum mechanics, and that time evolution (in the Krein space of states) is unitary if and only if Hamiltonian is $J$-Hermitian (or equivalently $PT$-symmetric). An issue with unitarity comes up when time evolution is considered in the Hilbert space of states rather than in the Krein space of states. The author offers possible scenarios with this issue.

arXiv:1312.7738Dec 30, 20132 citations

Time symmetry and interpretation of quantum mechanics

O. Costa de Beauregard

Foundations of PhysicsOct 1, 197667 citations

Update: A time-independent

n\times n

PT-symmetric (and symmetric) Hamiltonian is diagonalizable since it has all distinct real eigenvalues and the resulting diagonal matrix is a real symmetric matrix. The diagonalization results an isometry so there shouldn't be any issue with unitarity and unfortunately this very elementary mathematical fact somehow did not draw the authors' attention. However, PT-symmetric quantum mechanics is not out of trouble. For time-dependent PT-symmetric (and symmetric) Hamiltonians (even

2\times 2

ones) the authors observed that there is a violation of unitarity. Moreover, the first named author showed in his recent article arXiv:1312.7738 that PT-symmetric quantum mechanics is indeed a certain kind of Hermitian quantum mechanics and that in order for time-evolution to be unitary with respect to

J

-inner product (one that gives rise to a Hilbert space structure on the space of state functions), the potential energy operator

V(x)

must be real. This means that those complex PT-symmetric Hamiltonians that have been studied by physicists are unfortunately unphysical. The first named author discussed in a subsequent article arXiv:1401.5149 that while finite-state PT-symmetric quantum mechanics with time-independent Hamiltonians is not physically any different from Hermitian quantum mechanics, PT-symmetric quantum mechanics exhibits a distinctive symmetry from that of Hermitian quantum mechanics.

Time evolution and adiabatic approximation in

PT

-symmetric quantum mechanics

In this paper, we discuss time evolution and adiabatic approximation in

PT

-symmetric quantum mechanics. we give the time evolving equation for a class of

PT

-symmetric Hamiltonians and some conditions of the adiabatic approximation for the class of

PT

-symmetric Hamiltonians.

PT

-symmetric quantum mechanics is a Hermitian quantum mechanics

The author discusses a different kind of Hermitian quantum mechanics, called

J

-Hermitian quantum mechanics. He shows that

PT

-symmetric quantum mechanics is indeed

J

-Hermitian quantum mechanics, and that time evolution (in the Krein space of states) is unitary if and only if Hamiltonian is

J

-Hermitian (or equivalently

PT

-symmetric). An issue with unitarity comes up when time evolution is considered in the Hilbert space of states rather than in the Krein space of states. The author offers possible scenarios with this issue.