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Philip D Mannheim - One of the best experts on this subject based on the ideXlab platform.
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goldstone bosons and the englert brout higgs mechanism in non hermitian theories
Physical Review D, 2019Co-Authors: Philip D MannheimAbstract:In recent work Alexandre, Ellis, Millington and Seynaeve have extended the Goldstone theorem to non-Hermitian Hamiltonians that possess a discrete antilinear symmetry such as $PT$. They restricted their discussion to those realizations of antilinear symmetry in which all the energy eigenvalues of the Hamiltonian are real. Here we extend the discussion to the two other realizations possible with antilinear symmetry, namely energies in complex conjugate pairs or Jordan-Block Hamiltonians that are not diagonalizable at all. In particular, we show that under certain circumstances it is possible for the Goldstone boson mode itself to be one of the zero-norm states that are characteristic of Jordan-Block Hamiltonians. While we discuss the same model as Alexandre et al. our treatment is quite different, though their main conclusion that one can have Goldstone bosons in the non-Hermitian case remains intact. In their paper Alexandre et al. presented a variational procedure for the action in which the surface term played an explicit role, to thus suggest that one has to use such a procedure in order to establish the Goldstone theorem in the non-Hermitian case. However, by taking certain fields that they took to be Hermitian to actually either be anti-Hermitian or be made so by a similarity transformation, we show that we are then able to obtain a Goldstone boson using a completely standard variational procedure. Since we use a standard variational procedure we can readily extend our analysis to a continuous local symmetry by introducing a gauge boson. We show that when we do this the gauge boson acquires a non-zero mass by the Higgs mechanism in all realizations of the antilinear symmetry, except the one where the Goldstone boson itself has zero norm, in which case, and despite the fact that the continuous local symmetry has been spontaneously broken, the gauge boson remains massless.
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astrophysical evidence for the non hermitian but pt symmetric hamiltonian of conformal gravity
Protein Science, 2013Co-Authors: Philip D MannheimAbstract:In this review we discuss the connection between two seemingly disparate topics, macroscopic gravity on astrophysical scales and Hamiltonians that are not Hermitian but PT symmetric on microscopic ones. In particular we show that the quantum-mechanical unitarity problem of the fourth-order derivative conformal gravity theory is resolved by recognizing that the scalar product appropriate to the theory is not the Dirac norm associated with a Hermitian Hamiltonian but is instead the norm associated with a non-Hermitian but PT-symmetric Hamiltonian. Moreover, the fourth-order theory Hamiltonian is not only not Hermitian, it is not even diagonalizable, being of Jordan-Block form. With PT symmetry we establish that conformal gravity is consistent at the quantum-mechanical level. In consequence, we can apply the theory to data, to find that the theory is capable of naturally accounting for the systematics of the rotation curves of a large and varied sample of 138 spiral galaxies without any need for dark matter. The success of the fits provides evidence for the relevance of non-diagonalizable but PT-symmetric Hamiltonians to physics.
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exactly solvable pt symmetric hamiltonian having no hermitian counterpart
Physical Review D, 2008Co-Authors: Carl M Bender, Philip D MannheimAbstract:In a recent paper Bender and Mannheim showed that the unequal-frequency fourth-order derivative Pais-Uhlenbeck oscillator model has a realization in which the energy eigenvalues are real and bounded below, the Hilbert-space inner product is positive definite, and time evolution is unitary. Central to that analysis was the recognition that the Hamiltonian H{sub PU} of the model is PT symmetric. This Hamiltonian was mapped to a conventional Dirac-Hermitian Hamiltonian via a similarity transformation whose form was found exactly. The present paper explores the equal-frequency limit of the same model. It is shown that in this limit the similarity transform that was used for the unequal-frequency case becomes singular and that H{sub PU} becomes a Jordan-Block operator, which is nondiagonalizable and has fewer energy eigenstates than eigenvalues. Such a Hamiltonian has no Hermitian counterpart. Thus, the equal-frequency PT theory emerges as a distinct realization of quantum mechanics. The quantum mechanics associated with this Jordan-Block Hamiltonian can be treated exactly. It is shown that the Hilbert space is complete with a set of nonstationary solutions to the Schroedinger equation replacing the missing stationary ones. These nonstationary states are needed to establish that the Jordan-Block Hamiltonian of the equal-frequency Pais-Uhlenbeck model generates unitary timemore » evolution.« less
Pajitnov Andrei - One of the best experts on this subject based on the ideXlab platform.
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Twisted monodromy homomorphisms and Massey products
HAL CCSD, 2019Co-Authors: Pajitnov AndreiAbstract:16 pagesLet $\phi: M\to M$ be a diffeomorphism of a $C^\infty$ compact connected manifold, and $X$ its mapping torus. There is a natural fibration $p:X\to S^1$, denote by $\xi\in H^1(X, \mathbb{Z})$ the corresponding cohomology class. Let $\rho:\pi_1(X)\to GL(n,\mathbb{C})$ be a representation, denote by $H^*(X,\rho)$ the corresponding twisted cohomology of $X$. Denote by $\rho_0$ the restriction of $\rho$ to $\pi_1(M)$, and by $\rho^*_0$ the antirepresentation conjugate to $\rho_0$. We construct from these data an automorphism of the group $H_*(M,\rho^*_0)$, that we call the twisted monodromy homomorphism $\phi_*$. The aim of the present work is to establish a relation between Massey products in $H^*(X,\rho)$ and Jordan Blocks of $\phi_*$. Given a non-zero complex number $\lambda$ define a representation $\rho_\lambda:\pi_1(X)\to GL(n,\mathbb{C})$ as follows: $\rho_\lambda(g)=\lambda^{\xi(g)}\cdot\rho(g)$. Denote by $J_k(\phi_*, \lambda)$ the maximal size of a Jordan Block of eigenvalue $\lambda$ of the automorphism $\phi_*$ in the homology of degree $k$. The main result of the paper says that $J_k(\phi_*, \lambda)$ is equal to the maximal length of a non-zero Massey product of the form $\langle \xi, \ldots , \xi, x\rangle$ where $x\in H^k(X,\rho)$ (here the length means the number of entries of $\xi$). In particular, $\phi_*$ is diagonalizable, if a suitable formality condition holds for the manifold $X$. This is the case if $X$ a compact K\"ahler manifold and $\rho$ is a semisimple representation. The proof of the main theorem is based on the fact that the above Massey products can be identified with differentials in a Massey spectral sequence, which in turn can be explicitly computed in terms of the Jordan normal form of $\phi_*$
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Massey products in mapping tori
HAL CCSD, 2019Co-Authors: Pajitnov AndreiAbstract:Revised for publication in European Journal of Mathematics. Published online 10 November 2016Let $\phi: M\to M$ be a diffeomorphism of a $C^\infty$ compact connected manifold, and $X$ its mapping torus. There is a natural fibration $p:X\to S^1$, denote by $\xi\in H^1(X, \mathbb{Z})$ the corresponding cohomology class. Let $\lambda\in \mathbb{Z}^*$. Consider the endomorphism $\phi_k^*$ induced by $\phi$ in the cohomology of $M$ of degree $k$, and denote by $J_k(\lambda)$ the maximal size of its Jordan Block of eigenvalue $\lambda$. Define a representation $\rho_\lambda : \pi_1(X)\to\mathbb{C}^*$ by $\rho_\lambda (g) = \lambda^{p_*(g)}.$ Let $H^*(X,\rho_\lambda)$ be the corresponding twisted cohomology of $X$. We prove that $J_k(\lambda)$ is equal to the maximal length of a non-zero Massey product of the form $\langle \xi, \ldots , \xi, a\rangle$ where $a\in H^k(X,\rho_\lambda)$ (here the length means the number of entries of $\xi$). In particular, if $X$ is a strongly formal space (e.g. a K\"ahler manifold) then all the Jordan Blocks of $\phi_k^*$ are of size 1. If $X$ is a formal space, then all the Jordan Blocks of eigenvalue 1 are of size 1. This leads to a simple construction of formal but not strongly formal mapping tori. The proof of the main theorem is based on the fact that the Massey products of the above form can be identified with differentials in a Massey spectral sequence, which in turn can be explicitly computed in terms of the Jordan normal form of $\phi^*$
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Twisted monodromy homomorphisms and Massey products
2017Co-Authors: Pajitnov AndreiAbstract:Let $\phi: M\to M$ be a diffeomorphism of a $C^\infty$ compact connected manifold, and $X$ its mapping torus. There is a natural fibration $p:X\to S^1$, denote by $\xi\in H^1(X, \mathbb{Z})$ the corresponding cohomology class. Let $\rho:\pi_1(X)\to GL(n,\mathbb{C})$ be a representation, denote by $H^*(X,\rho)$ the corresponding twisted cohomology of $X$. Denote by $\rho_0$ the restriction of $\rho$ to $\pi_1(M)$, and by $\rho^*_0$ the antirepresentation conjugate to $\rho_0$. We construct from these data an automorphism of the group $H_*(M,\rho^*_0)$, that we call the twisted monodromy homomorphism $\phi_*$. The aim of the present work is to establish a relation between Massey products in $H^*(X,\rho)$ and Jordan Blocks of $\phi_*$. Given a non-zero complex number $\lambda$ define a representation $\rho_\lambda:\pi_1(X)\to GL(n,\mathbb{C})$ as follows: $\rho_\lambda(g)=\lambda^{\xi(g)}\cdot\rho(g)$. Denote by $J_k(\phi_*, \lambda)$ the maximal size of a Jordan Block of eigenvalue $\lambda$ of the automorphism $\phi_*$ in the homology of degree $k$. The main result of the paper says that $J_k(\phi_*, \lambda)$ is equal to the maximal length of a non-zero Massey product of the form $\langle \xi, \ldots , \xi, x\rangle$ where $x\in H^k(X,\rho)$ (here the length means the number of entries of $\xi$). In particular, $\phi_*$ is diagonalizable, if a suitable formality condition holds for the manifold $X$. This is the case if $X$ a compact K\"ahler manifold and $\rho$ is a semisimple representation. The proof of the main theorem is based on the fact that the above Massey products can be identified with differentials in a Massey spectral sequence, which in turn can be explicitly computed in terms of the Jordan normal form of $\phi_*$.Comment: 16 page
Yong Chen - One of the best experts on this subject based on the ideXlab platform.
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Jordan decomposition and geometric multiplicity for a class of non symmetric ornstein uhlenbeck operators
Advances in Difference Equations, 2014Co-Authors: Jiying Wang, Yong ChenAbstract:In this paper, we calculate the Jordan decomposition for a class of non-symmetric Ornstein-Uhlenbeck operators with the drift coefficient matrix, being a Jordan Block, and the diffusion coefficient matrix, being the identity multiplying a constant. For the 2-dimensional case, we present all the general eigenfunctions by mathematical induction. For the 3-dimensional case, we divide the calculation of the Jordan decomposition into three steps. The key step is to do the canonical projection onto the homogeneous Hermite polynomials, and then use the theory of systems of linear equations. Finally, we get the geometric multiplicity of the eigenvalue of the Ornstein-Uhlenbeck operator.
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on the Jordan decomposition for a class of non symmetric ornstein uhlenbeck operators
arXiv: Probability, 2012Co-Authors: Yong Chen, Ying LiAbstract:In this paper, we calculate the Jordan decomposition (or say, the Jordan canonical form) for a class of non-symmetric Ornstein-Uhlenbeck operators with the drift coefficient matrix being a Jordan Block and the diffusion coefficient matrix being identity multiplying a constant. For the 2-dimensional case, we present all the general eigenfunctions by the induction. For the 3-dimensional case, we divide the calculating of the Jordan decomposition into several steps (the key step is to do the canonical projection onto the homogeneous Hermite polynomials, next we use the theory of systems of linear equations). As a by-pass product, we get the geometric multiplicity of the eigenvalue of the Ornstein-Uhlenbeck operator.
Kurt J. Reinschke - One of the best experts on this subject based on the ideXlab platform.
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digraph based determination of Jordan Block size structure of singular matrix pencils
Linear Algebra and its Applications, 1998Co-Authors: Klaus Röbenack, Kurt J. ReinschkeAbstract:Abstract The generic Jordan Block sizes corresponding to multiple characteristic roots at zero and at infinity of a singular matrix pencil will be determined graph-theoretically. An application of this technique to detect certain controllability properties of linear time-invariant differential algebraic equations is discussed.
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graph theoretically determined Jordan Block size structure of regular matrix pencils
Linear Algebra and its Applications, 1997Co-Authors: Klaus Röbenack, Kurt J. ReinschkeAbstract:Abstract The authors investigate the sizes of Jordan Blocks of regular matrix pencils by means of a one-to-one correspondence between a matrix pencil ( λE + μA ) and a weighted digraph G ( E , A ). Based on the relationship between determinantal divisors of a pencil and spanning-cycle families of the associated digraph G ( E , A ), the Jordan-Block-size structure is determined graph-theoretically. For classes of structurally equivalent matrix pencils defined by a pair of structure matrices [ E , A ], the generic Jordan Block sizes corresponding to the characteristic roots at zero and at infinity can be obtained from the unweighted digraph G ([ E ], [ A ]). Eigenvalues of matrices are discussed as special cases. A nontrivial mechanical example illustrates the procedure.
Carl M Bender - One of the best experts on this subject based on the ideXlab platform.
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exactly solvable pt symmetric hamiltonian having no hermitian counterpart
Physical Review D, 2008Co-Authors: Carl M Bender, Philip D MannheimAbstract:In a recent paper Bender and Mannheim showed that the unequal-frequency fourth-order derivative Pais-Uhlenbeck oscillator model has a realization in which the energy eigenvalues are real and bounded below, the Hilbert-space inner product is positive definite, and time evolution is unitary. Central to that analysis was the recognition that the Hamiltonian H{sub PU} of the model is PT symmetric. This Hamiltonian was mapped to a conventional Dirac-Hermitian Hamiltonian via a similarity transformation whose form was found exactly. The present paper explores the equal-frequency limit of the same model. It is shown that in this limit the similarity transform that was used for the unequal-frequency case becomes singular and that H{sub PU} becomes a Jordan-Block operator, which is nondiagonalizable and has fewer energy eigenstates than eigenvalues. Such a Hamiltonian has no Hermitian counterpart. Thus, the equal-frequency PT theory emerges as a distinct realization of quantum mechanics. The quantum mechanics associated with this Jordan-Block Hamiltonian can be treated exactly. It is shown that the Hilbert space is complete with a set of nonstationary solutions to the Schroedinger equation replacing the missing stationary ones. These nonstationary states are needed to establish that the Jordan-Block Hamiltonian of the equal-frequency Pais-Uhlenbeck model generates unitary timemore » evolution.« less
