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We propose a history state formalism for a Dirac particle. By introducing a reference quantum clock system it is first shown that Diracs equation can be derived by enforcing a timeless Wheeler-DeWitt-like equation for a global state. The Hilbert space of the whole system constitutes a unitary representation of the Lorentz group with respect to a properly defined invariant product, and the proper normalization of global states directly ensures standard Diracs norm. Moreover, by introducing a second quantum clock, the previous invariant product emerges naturally from a generalized continuity equation. The invariant parameter $tau$ associated with this second clock labels history states for different particles, yielding an observable evolution in the case of an hypothetical superposition of different masses. Analytical expressions for both space-time density and electron-time entanglement are provided for two particular families of electrons states, the former including Pryce localized particles.
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We present a covariant quantum formalism for scalar particles based on an enlarged Hilbert space. The particular physical theory can be introduced through a timeless Wheeler DeWitt-like equation, whose projection onto four-dimensional coordinates leads to the Klein Gordon equation. The standard quantum mechanical product in the enlarged space, which is invariant and positive definite, implies the usual Klein Gordon product when applied to its eigenstates. Moreover, the standard three-dimensional invariant measure emerges naturally from the flat measure in four dimensions when mass eigenstates are considered, allowing a rigorous identification between definite mass history states and the standard Wigner representation. Connections with the free propagator of scalar field theory and localized states are subsequently derived. The formalism also allows the superposition of different theories and remains valid in the presence of a fixed external field, revealing special orthogonality relations. Other details such as extended identities for the current density, the quantization of parameterized theories and the nonrelativistic limit, with its connection to the Page and Wootters formalism, are discussed. A related consistent second quantization formulation is also introduced.
By a series of simple examples, we illustrate how the lack of mathematical concern can readily lead to surprising mathematical contradictions in wave mechanics. The basic mathematical notions allowing for a precise formulation of the theory are then summarized and it is shown how they lead to an elucidation and deeper understanding of the aforementioned problems. After stressing the equivalence between wave mechanics and the other formulations of quantum mechanics, i.e. matrix mechanics and Diracs abstract Hilbert space formulation, we devote the second part of our paper to the latter approach: we discuss the problems and shortcomings of this formalism as well as those of the bra and ket notation introduced by Dirac in this context. In conclusion, we indicate how all of these problems can be solved or at least avoided.
A neo-classical relativistic mechanics model is presented where the spin of an electron is a natural part of its space-time path. The fourth-order equation of motion corresponds to the same Lagrangian function in proper time as in special relativity except for an additional spin energy term. The total motion can be decomposed into a sum of a local circular motion giving the spin and a global motion of the spin center, each being governed by a second-order differential equation. The local spin motion corresponds to Schrodingers zitterbewegung and is a perpetual motion; it produces magnetic and electric dipoles through the Lorentz force on the electrons point charge. The global motion is sub-luminal and described by Newtons second law in proper time, the time for a clock fixed at the spin center, where the inertia to acceleration resides. The total motion occurs at the speed of light c, consistent with the eigenvalues of Diracs velocity operators having magnitude c. A spin tensor is introduced that is the angular momentum of the electrons total motion about its spin center. The fundamental equations of motion expressed using this tensor are identical to those of the Barut-Zanghi theory; they can be expressed in an equivalent form using the same operators as in Diracs theory for the electron but applied to a state function of proper time satisfying a Dirac-Schrodinger spinor equation. This state function produces a neo-classical wave function that satisfies Diracs relativistic wave equation for the free electron when the Lorentz transformation is used to express proper time in terms of an observers space-time coordinates. In summary, the theory provides a hidden-variable model for spin that leads to Diracs relativistic wave equation and explains the electrons moment coupling to an electro-magnetic field, albeit with a magnetic moment that is one half of that in Diracs theory.
We present a method to apply the well-known matrix product state (MPS) formalism to partially separable states in solid state systems. The computational effort of our method is equal to the effort of the standard density matrix renormalisation group (DMRG) algorithm. Consequently, it is applicable to all usually considered condensed matter systems where the DMRG algorithm is successful. We also show in exemplary cases, that polymerisation properties of ground states are closely connected to properties of partial separability, even if the ground state itself is not partially separable.
We prove a `resilience version of Diracs theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $varepsilon>0$, a.a.s. the following holds: let $G$ be any subgraph of the random $n$-vertex $d$-regular graph $G_{n,d}$ with minimum degree at least $(1/2+varepsilon)d$. Then $G$ is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that $d$ is large cannot be omitted, and secondly, the minimum degree bound cannot be improved.
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