Dirac Equation Redux by Direct Quantization of the 4-Momentum Vector

Dirac equation (DE) is a cornerstone of quantum physics. We prove that direct quantization of the 4-momentum vector p with modulus 𝑚𝑐 (𝑚 is rest mass) yields a coordinate-free and manifestly covariant equation. In coordinate representation, this is equivalent to DE with spacetime frame vectors xμ replacing Dirac’s γμ -matrices. Remember that standard DE is not manifestly covariant. The two sets {xμ}, {γμ} obey to the same Clifford algebra. Adding an independent Hermitian vector x5 to the spacetime basis {xμ} allows to accommodate the momentum operator in a real vector space with a complex structure generated alone by vectors and multivectors. The real vector space arising from the action of the Clifford product onto the quintet {xμ , x5 } has dimension 32, the same as the equivalent real dimension for the space of Dirac matrices. x5 proves defining for the combined CPT symmetry, axial vs. polar vectors, left and right handed rotors & spinors, etc.; therefore, we name it reflector and {xμ , x5 } – a basis for spacetime-reflection (STR). The pentavector 𝐼 ≡ x05123 in STR substitutes the imaginary unit i. We develop the formalism by deriving all the essential results from the novel STR DE: conserved probability currents, symmetries, nonrelativistic approximation and spin 1/2 magnetic angular momentum. It will become clear that key symmetries follow more directly and with clearer geometric interpretation in STR than in the standard approach. In simple terms, we demonstrate how Dirac matrices are a redundant representation of spacetime-reflection directors.

Geometry on the variety of subspaces with positive definite form: geometry of the Grassmann spaces over generalized hyperbolic spaces

We consider Grassmann structures defined on the family consisting of subspaces on which a given nondegenerate bilinear form defined on a real vector space is positive definite. One may call such structures Grassmann spaces over generalized hyperbolic spaces. We show that the underlying (generalized) hyperbolic space can be recovered in terms of its Grassmannian, and the underlying projective space (equipped with respective associated polarity) can be recovered in terms of the generalized hyperbolic space defined over it.

Criteria for periodicity and an application to elliptic functions

Let P and Q be relatively prime integers greater than 1, and let f be a real valued discretely supported function on a finite dimensional real vector space V. We prove that if $f_{P}(x)=f(Px)-f(x)$ and $f_{Q}(x)=f(Qx)-f(x)$ are both $\Lambda $ -periodic for some lattice $\Lambda \subset V$ , then so is f (up to a modification at $0$ ). This result is used to prove a theorem on the arithmetic of elliptic function fields. In the last section, we discuss the higher rank analogue of this theorem and explain why it fails in rank 2. A full discussion of the higher rank case will appear in a forthcoming work.

Why can states and measurement outcomes be represented as vectors?

It is shown how, given a "probability data table" for a quantum or classical system, the representation of states and measurement outcomes as vectors in a real vector space follows in a natural way. Some properties of the resulting sets of these vectors are discussed, as well as some connexions with the quantum-mechanical formalism.

Cross-Ratio in Real Vector Space

Summary Using Mizar [1], in the context of a real vector space, we introduce the concept of affine ratio of three aligned points (see [5]). It is also equivalent to the notion of “Mesure algèbrique”1, to the opposite of the notion of Teilverhältnis2 or to the opposite of the ordered length-ratio [9]. In the second part, we introduce the classic notion of “cross-ratio” of 4 points aligned in a real vector space. Finally, we show that if the real vector space is the real line, the notion corresponds to the classical notion3 [9]: The cross-ratio of a quadruple of distinct points on the real line with coordinates x1, x2, x3, x4 is given by: $$({x_1},{x_2};{x_3},{x_4}) = {{{x_3} - {x_1}} \over {{x_3} - {x_2}}}.{{{x_4} - {x_2}} \over {{x_4} - {x_1}}}$$ In the Mizar Mathematical Library, the vector spaces were first defined by Kusak, Leonczuk and Muzalewski in the article [6], while the actual real vector space was defined by Trybulec [10] and the complex vector space was defined by Endou [4]. Nakasho and Shidama have developed a solution to explore the notions introduced by different authors4 [7]. The definitions can be directly linked in the HTMLized version of the Mizar library5. The study of the cross-ratio will continue within the framework of the Klein- Beltrami model [2], [3]. For a generalized cross-ratio, see Papadopoulos [8].

Hyperplanes That Intersect Each Ray of a Cone Once and a Banach Space Counterexample

Suppose C is a cone contained in real vector space V. When does V contain a hyperplane H that intersects each of the 0-rays in C∖{0} exactly once? We build on results found in Aliprantis, Tourky, and Klee Jr.’s work to give a partial answer to this question. We also present an example of a salient, closed Banach space cone C for which there does not exist a hyperplane that intersects each 0-ray in C∖{0} exactly once.