The Scattering Algebra of Physical Space: Squared Massive Constructive Amplitudes

ABSTRACTAn approach aimed to connect configuration and algebraic spaces is discussed. This approach emerges as the need to translate a vibrational description in configuration space to an algebraic representation based on unitary dynamical algebras, where a straightforward connection does not exist e.g. the vibron model case. Our method is based on the mapping of the algebraic to configuration states, a premise that allows arbitrary operators in configuration to be expanded in terms of generators of the dynamical algebra. The coefficients are determined through a minimisation procedure and given in terms of matrix elements defined in configuration space. We apply the general formalism to the Morse potential, representing anharmonic vibrations in a molecule, as a benchmark case where a dynamical symmetry exits, and to the symmetric double-Morse potential, representing vibrations that can tunnel through a potential barrier, as an example in which a dynamical symmetry is not present. We discuss how the tunnelling effect in the double Morse can be described very simply in the su(2) algebraic representation, taking the ammonia inversion vibrational spectrum as an example.

In this chapter, we study the vector algebra of 3-dimensional space. The term “algebra” is meant here in its mathematical sense, so that in addition to the usual vector-space manipulations, an associative multiplication of vectors is required. Relatively simple considerations lead us to what is called the geometric algebra(or Clifford algebra)of 3dimensional space, also known as the Pauli algebra. The standard matrix representation of this algebra replaces basis vectors by Pauli spin matrices (and hence the name “Pauli algebra”), but specific representations encumber the mathematics with unnecessary baggage; it is usually simpler to work directly in the algebra in component-free notation without reference to any matrices.

A finite-range electromagnetic (EM) theory containing both electric and magnetic charges constructed using two vector potentials Aμ and Zμ is formulated in the spacetime algebra (STA) and in the algebra of the three-dimensional physical space (APS) formalisms. Lorentz, local gauge and EM duality invariances are discussed in detail in the APS formalism. Moreover, considerations about signature and dimensionality of spacetime are discussed. Finally, the two formulations are compared. STA and APS are equally powerful in formulating our model, but the presence of a global commuting unit pseudoscalar in the APS formulation and the consequent possibility of providing a geometric interpretation for the imaginary unit employed throughout physics lead us to prefer the APS approach.

A complex and, equivalently, hyperbolic extension of the algebra of physical space (APS) is discussed that allows one to distinguish space-time vectors from paravectors of APS, while preserving the natural origin of the Minkowski space-time metric. The CAPS formalism is Lorentz covariant and gives expression to persistent vectors in physical space as time-like planes in space-time. Commuting projectors $${P_{\pm} = \frac{1}{2} (1 \pm h)}$$ project CAPS onto two-sided ideals, one of which is APS. CAPS has the same dimension as the space-time algebra (STA) if both are considered real algebras, and it distinguishes covariant roles of elements, as does STA. Its structure, however, is closer to APS, with a volume element that belongs to the center of the algebra and a simple relation between space-times of opposite signature. Furthermore, CAPS, unlike STA, distinguishes point-like space-time inversion of a Dirac spinor from a physical rotation. To illustrate its use, CAPS is applied to the Dirac equation and to the fundamental symmetry transformations of the equation and Dirac spinors. The physical interpretations of both the equation and the spinor are clarified, and it is seen that the space-time frame $${\{\gamma_{\mu}\}}$$ arises fully from relative vectors and does not imply the existence of an absolute space-time frame.

The algebra of physical space (APS) is a name for the Clifford or geometric algebra, which can be closely associated with the geometry of special relativity and relativistic spacetime. For example, the Dirac Hamiltonian can be presented as the scalar product of the electron's four‐momentum and Dirac's four‐vector of gamma matrices, $({\gamma_0},\vec \gamma)$, the latter of which is a Clifford algebra. We show here that a geometric spacetime or four‐space solution of Dirac's equation conforms to the principles of APS, an early example of which is Schroedinger's solution of Dirac's equation for a free electron, which exhibits Zitterbewegung. In a four‐space solution the spacetime coordinates, $\vec r$ and the scaled time ct, are treated on an equal footing as physical observables to avoid any suggestion of a preferred frame of reference. The geometric spacetime theory is studied here for the Coulomb problem. The positive‐energy spectrum of states is found to be identical within numerical error to that of standard Dirac's theory, but the wave function exhibits Zitterbewegung. It is shown analytically how the geometric spacetime solution can be reduced to the standard solution of Dirac's equation, in which Zitterbewegung is absent. The rigor of APS and of its conforming geometric spacetime solution provide strong support for further investigation into the physical interpretation of the geometric spacetime Dirac's wave function and Zitterbewegung. © 2012 Wiley Periodicals, Inc.

The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term in the usual Dirac factorization of the Klein–Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.

Elko spinors are eigenspinors of the charge conjugation operator. In this work we use the Clifford algebra of the physical space in order to formulate the theory of Elko spinors and use a procedure analog to Ryder’s derivation of Dirac equation to come up with an equation for Elko spinor fields. Unlike other works in the literature where an equation for Elko spinor fields has been studied, in this work we obtain a first order differential equation for Elko spinor fields, which resembles but is different from the Dirac equation.

Of the various formalisms developed to treat relativistic phenomena, those based on Clifford's geometric algebra are especially well adapted for clear geometric interpretations and computational efficiency. Here we study relationships between formulations of special relativity in the spacetime algebra (STA) Cl{1,3} of Minkowski space, and in the algebra of physical space (APS)Cl{3}. STA lends itself to an absolute formulation of relativity, in which paths, fields, and other physical properties have observer-independent representations. Descriptions in APS are related by a one-to-one mapping of elements from APS to the even subalgebra STA+ of STA. With this mapping, reversion in APS corresponds to hermitian conjugation in STA. The elements of STA+ are all that is needed to calculate physically measurable quantities because only they entail the observer dependence inherent in any physical measurement. As a consequence, every relativistic physical process that can be modeled in STA also has a representation in APS, and vice versa. In the presence of two or more inertial observers, two versions of APS present themselves. In the absolute version, both the mapping to STA+ and hermitian conjugation are observer dependent, and the proper basis vectors are persistent vectors that sweep out timelike planes. In the relative version, the mapping and hermitian conjugation are then the same for all observers. Relative APS gives a covariant representation of relativistic physics with spacetime multivectors represented by multiparavectors. We relate the two versions of APS as consistent models within the same algebra.

We translate the Dirac equation into the Clifford algebra of physical space. We study the second-order equation, the relativistic invariance, the gauge invariance, the Lagrangian density and the tensors of the Dirac theory. Next we completely solve, by separation of variables, the Dirac equation for the hydrogen atom. The classical solutions have vanishing invariants and we calculate some linear combinations of the classical solutions with nonvanishing invariants. These solutions may be the linear approximations for a nonlinear equation previously studied.

Although intrinsic spin is usually viewed as a purely quantum property with no classical analog, we present evidence that fermion spin has a classical origin rooted in the geometry of three-dimensional physical space. Our approach to the quantum/classical interface is based on a formulation of relativistic classical mechanics that uses spinors. Spinors and projectors arise naturally in the Clifford’s geometric algebra of physical space and not only provide powerful tools in classical electrodynamics, but also reproduce a number of quantum results. We show in particular that many properties of elementary fermions, as spin-1/2 particles, are obtained classically and relate spin, the associated g-factor, its coupling to an external magnetic field, Zitter-bewegung, and de Broglie waves. Spinors are also amplitudes that can undergo quantum-like interference. The relationship of spin and geometry is further strengthened by the fact that physical space and its geometric algebra can be derived from fermion annihilation and creation operators. The approach is important because of the insights it provides about spin and quantum phenomena more generally.

Clifford’s geometric algebra, in particular the algebra of physical space (APS), lubricates the paradigm shifts from the Newtonian worldview to the post-Newtonian theories of relativity and quantum mechanics. APS is an algebra of vectors in physical space, and its linear subspaces include a 4-dimensional space of paravectors (scalars plus vectors). The metric of the latter has the pseudo-Euclidean form of Minkowski spacetime, with which APS facilitates the transition from Newtonian mechanics to relativity without the need of tensors or matrices. APS also provides tools, such as spinors and projectors, for solving classical problems and for smoothing the transition to quantum theory. This lecture concentrates on paravectors and applications to relativity and electromagnetic waves. A following lecture will extend the treatment to the quantum/classical interface.

The exceptional Lie group E8 plays a prominent role in both mathematics and theoretical physics. It is the largest symmetry group associated with the most general possible normed division algebra, namely, that of the non-associative real octonions, which—thanks to their non-associativity—form the only possible closed set of spinors (or rotors) that can parallelize the 7-sphere. By contrast, here we show how a similar 7-sphere also arises naturally from the algebraic interplay of the graded Euclidean primitives, such as points, lines, planes and volumes, which characterize the three-dimensional conformal geometry of the ambient physical space, set within its eight-dimensional Clifford-algebraic representation. Remarkably, the resulting algebra remains associative, and allows us to understand the origins and strengths of all quantum correlations locally, in terms of the geometry of the compactified physical space, namely, that of a quaternionic 3-sphere, S3, with S7 being its algebraic representation space. Every quantum correlation can thus be understood as a correlation among a set of points of this S7, computed using manifestly local spinors within S3, thereby extending the stringent bounds of ±2 set by Bell inequalities to the bounds of on the strengths of all possible strong correlations, in the same quantitatively precise manner as that predicted within quantum mechanics. The resulting geometrical framework thus overcomes Bell’s theorem by producing a strictly deterministic and realistic framework that allows a locally causal understanding of all quantum correlations, without requiring either remote contextuality or backward causation. We demonstrate this by first proving a general theorem concerning the geometrical origins of the correlations predicted by arbitrarily entangled quantum states, and then reproducing the correlations predicted by the EPR-Bohm and the GHZ states. The raison d’être of strong correlations turns out to be the Möbius-like twists in the Hopf bundles of S3 and S7.

Clifford’s geometric algebra, in particular the algebra of physical space (APS), provides a new relativistic approach to the Quantum/Classical interface. It describes classical relativistic dynamics in quantum terms: spinor amplitudes with projectors describe classical motion and satisfy the Dirac equation of relativistic quantum theory. Some basic properties such as the spin-1/2 nature of elementary systems are seen to be a simple result of the geometry of physical space. The nature of “quantum reality” is constrained: the pure state of any single fermion is fully polarized and determines an exact spin direction, but an entangled pair can be unpolarized. Measurements can form or break entanglement, or they can transfer it between particle pairs. Quantum “weirdness” can arise from the need to use amplitudes of entangled systems.

This paper is about using Clifford Algebra to position a gimbal test fixture that is used during infrared laser (IR) pointing system testing. The Clifford Algebra Cl <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3, 0</sub> specifically known as the Algebra of Physical Space (APS) is useful as a unifying mathematical frame work in physics that can describe classical physics, special relativity, general relativity, electro-dynamics, and quantum mechanics in a consistent and concise way. This paper seeks to demonstrate the use of APS in controlling a simple 2 degree of freedom (DOF) gimbal system that would be useful for testing a 2DOF unit under test (UUT). The goal of the test is to rotate the UUT to a specific test position and point the UUT towards optical test equipment. This requires the accurate and reliable transformation of multiple reference frames and coordinate systems. The reason why APS is useful for rotations is because the even sub-algebra of APS is homomorphic to Hamilton's quaternion algebra. Clifford Algebra Cl <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3, 0</sub> can be implemented numerically on a computer using the matrix representation of the Pauli algebra of 2×2 complex matrices. This paper will show the relationship between APS, quaternions, and Pauli matrices. The paper will also cover all of the basic operations of forming paravectors, rotors/eigenspinors, and converting to and from the matrix form. The paper will cover transformation and inverse transformations of reference frames and vectors in our example system.

Chapter One - Invariant Quantum Wave Equations and Double Space-Time