A real expectation value of the time-dependent non-Hermitian Hamiltonians

With the aim to solve the time-dependent Schr ̈odinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent PT-symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the C(t)PT -inner product, where C(t) is a obtained from the charge conjugation operator C through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the C(t)PT normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.

Method of reconfiguring industrial facilities using the apparatus of piecewise linear aggregates

A method has been developed for structural and functional reconfiguration of industrial facilities based on piecewise linear aggregates, with ranking elements according to the importance of fulfilling their target function. The method allows for the conjugation of elements of complex technical objects using the conjugation operator. An example of the application of the proposed method is given. Keywords: reconfiguration, industrial facility, system, piecewise linear aggregates, synthesis of elements. [email protected], [email protected]

The Dirac Sea, T and C Symmetry Breaking, and the Spinor Vacuum of the Universe

We have developed a quantum field theory of spinors based on the algebra of canonical anticommutation relations (CAR algebra) of Grassmann densities in the momentum space. We have proven the existence of two spinor vacua. Operators C and T transform the normal vacuum into an alternative one, which leads to the breaking of the C and T symmetries. The CPT is the real structure operator; it preserves the normal vacuum. We have proven that, in the theory of the Dirac Sea, the formula for the charge conjugation operator must contain an additional generalized Dirac conjugation operator.

Biquaternionic Dirac Equation Predicts Zero Mass for Majorana Fermions

A biquaternionic version of the Dirac Equation is introduced, with a procedure for converting four-component spinors to elements of the Pauli algebra. In this version, mass appears as a coefficient between the 4-gradient of a spinor and its image under an outer automorphism of the Pauli algebra. The charge conjugation operator takes a particulary simple form in this formulation and switches the sign of the mass coefficient, so that for a solution invariant under charge conjugation the mass has to equal zero. The multiple of the charge conjugation operator by the imaginary unit turns out to be a complex Lorentz transformation. It commutes with the outer automorphism, while the charge conjugation operator itself anticommutes with it, providing a second more algebraic proof of the main theorem. Considering the Majorana equation, it is shown that non-zero mass of its solution is imaginary.

Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots

The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov–Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies as well to classical knots and classical knot cobordisms. We give an alternate formulation for the Manturov definition [34] of Khovanov homology [25], [26] for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We use this to show that a large class of virtual knots with unit Jones polynomial is non-classical, proving a conjecture in [20] and [10]. We then discuss the implications of the maps induced in the aforementioned theory to the universal Frobenius algebra [27] for virtual knots. Next we show how one can apply the Karoubi envelope approach of Bar-Natan and Morrison [3] on abstract link diagrams [17] with cross cuts to construct the canonical generators of the Khovanov–Lee homology [30]. Using these canonical generators we derive a generalization of the Rasmussen invariant [39] for virtual knot cobordisms and generalize Rasmussen’s result on the slice genus for positive knots to the case of positive virtual knots. It should also be noted that this generalization of the Rasmussen invariant provides an easy to compute obstruction to knot cobordisms in [Formula: see text] in the sense of Turaev [42].

Self-interacting mass-dimension one fields for any spin

According to Ahluwalia and Grumiller, massive spin-half fields of mass-dimension one can be constructed using the eigenspinors of the charge-conjugation operator (Elko) as expansion coefficients. In this paper, we generalize their result by constructing quantum fields from higher-spin Elko. The kinematics of these fields are thoroughly investigated. Starting with the field operators, their propagators and Hamiltonians are derived. These fields satisfy the higher-spin generalization of the Klein–Gordon but not the Dirac equation. Independent of the spin, they are all of mass-dimension one and are thus endowed with renormalizable self-interactions. These fields violate Lorentz symmetry. The violation can be characterized by a non-Lorentz-covariant term that appears in the Elko spin-sums. This term provides a decomposition of the generalized higher-spin Dirac operator in the momentum space thus suggesting a possible connection between the mass-dimension one fields and the Lorentz-invariant fields.