Tuesday 12 June 2018

Handedness in Physical theories

Linear Operators and Oriented Parallelepiped


In Middle school we are asked to solve a pair of simultaneous equations perhaps with the motivation that those linear equations represent some crossing of trajectories in space and that in solving the set we are determining where an intersection takes place.

Such a set of equations: 5x- 2y= jx and -2x+2y=ky can be represented as a matrix equation. With two free parameters, j and k for two equations we have a suite of possible solutions for the different lambda. Recast in this matrix form our system of simultaneous linear equations is an eigenvalue problem with  eigenvalues 1 and 6. The solution to a pair of simultaneous equations thus describes a parametrised suite of straight lines.

The set describes straight lines with a variable gradient, m=(5-j)/2 or 2/(2-k). The simultaneous equation is configured as a matrix, A operating on a column vector, x. The eigenvalue problem asks for a pair of straight lines (from the infinite set of possible two straight lines intersecting at the origin), which are orthogonal to each other. That they are perpendicular means that the eigenvector solutions describing the lines are at right angles, thus linear independent forming a vectors basis.

Operationally, to find this linear independent set of eigenvectors that span the space of solutions we
take the determinant of the matrix A (denoted by |... |). As a set of solutions, all other possible solutions are mere combinations of these bases vectors. In our example the eigenvalues are 6 an 1 which tells us that the eigenvectors are (-1,-2) and (-2,1) or (4,-2) and (-2,-4). We can roughly think of matrices as operators (in a particular basis) that act on vectors, quite generally  rotating, stretching, compressing or displacing them in space. Eigen-vectors are special column vectors that when acted upon by such matrices merely react by stretching or compressing (mathinsight.org)  in length.

In Quantum (Linear) Mechanics these Linear operators give rise to the observables (vector) states that we see as the s and p shell eigenvalues of an atom. Linear Operators mark out the energies of the atom's states.

Geometrically what we are doing in looking for Eigenvectors, is to find the set of vectors that span our solution space - that is from which we can, for example, form a mutually orthogonal bases set of which all other solutions are a mere combination of such base factors.

We could envisage any number of such equations to solve and quite generally we could have a set of n simultaneous equations, with n unknowns.  We have abstracted our familiar world of the space of two-dimensional vectors to (possibly infinitely dimensional) solutions of polynomial equations. This is what mathematics is about: have a concrete conception identifying the consistent set of  rules that are used to manipulate these objects and generalise as far as you can consistently. The rules in this case are of Linearity: those that insist on Associativity, Distributivity (required for addition and multiplication of Real numbers) and the need to have a zero element amongst others. Apply then these self-consistent set of operating rules to new objects of interest and up open up the realm of functional analysis.

In the 3-d picture we are defining an oriented plane spanned by the bases vectors. That the two vectors are by definition directed line segments means that there is a face up and face down defined to that plane. To the right we have a plane spanned by the position vectors a and b. The vector cross product a x b is defined as a vector pointing in the direction of the thumb of the left hand if the plane support is spanned by the left hand's middle and fore-fingers. The scalar triple product is defined as:

The short-hand algebraic method to represent this determinant operation that captures its anti-symmetry involves an object called the Levi-Cevita (tensor density) symbol, denoted by epsilon. 
Our cross product term is then written as:

The determinant operation may be interpreted as the shearing of an oriented (hyper-)volume of a skewed cube obtained by applying a linear operator, A to a standard 3-(hyper)cube, Ax=lx. Theories that involve a determinant operation define a choice of orientation that as such are not symmetric under a Parity operation. Such volumes thus embody an orientation of the space. 

The archetypical cross-product vector is the angular momentum (spin), J. 
With an orientation it is known as an axial vector: an orientated line segment perpendicular to the (oriented) plane that describes the rotation of a body around an axis. 

A sense is chosen as the left-hand rule in which the thumb directs the axial vector while the middle and forefingers span the support. In picking out a preferred direction we will see, unlike for normal vectors, J remains invariant to an inversion of co-ordinate axis from x to -x. We can think of such vectors, J as generating symmetries and thus conservations laws of our system. They are also observables for the same reason.
Some examples of orientation ¨aware¨ systems are cited below.

Spinor Fermions, Vector Gauge Bosons and Chirality


A non-chiral (i.e. parity symmetric preserving) theory is called a vector theory. The terms chiral or vector derive from the types of invariant objects that arise from the Representation of the underlying theory's Group of symmetries. In this sense the familiar "vectors" of three-dimensional space are the objects that are invariant -staying the same -when the underlying basis (co-ordinate axis) set is rotated. That is, the vector expresses the equivalence or indistinguishability of the object under rotations.

Quantum Chromo Dynamics (QCD), the quantum field theory that describes the (non-linear) theory of the strong interaction binding together the quarks of a nucleon is an example of a vector theory since both left and right-handed chiralities of all the quarks appear in the theory, and they couple the same way. The electroweak theory as part of QED controlling radioactive decay is a chiral theory despite one of its invariant objects having both right- and left hands. The object being the mass-less neutrino is described by a so-called Weyl spinor that is invariant under the (double cover) of the Lorentz transformations of Einstein’s Special theory of Relativity.


The Snap Modes of Vibration of a Gravitational Wave


Just as an electromagnetic wave is an oscillation in the electric and magnetic fields that propagates at the speed of light, so is a gravitational wave is an oscillation in the gravitational field. To generate such waves asymmetric collapses or expansions of matter are required as symmetric collapses cancel out far-field gravity wave formation. Resulting gravitational waves propagate through space-time at the speed of light, distorting space-time as it passed through it.

As other force-carrying particles a gravitational wave has integral spin. Being a spin-2 particle with a quadrupole moment, the gravitational wave, passing through any point in space, would both stretch space in one direction and compress the space in the orthogonal direction.

Gravitational waves if they are second quantizable would be carried by gravitons. A graviton is the excitation in the boson-gravitational field, travelling (according to 2017 results form binary-Neutron star coalescence to 15 orders of magnitude in precision to effectively ) the speed of light. It is a spin-2 particle, the only one, which means that it somehow needs only spin half a revolution before it arrives in the same position. As other force-carrying particles it has integral spin. Being a spin-2 particle with a quadrupole moment, the gravitational wave, passing through any point in space, would both stretch space in one direction and compress the space in the orthogonal direction.

In terms of generating a gravitational wave we have the following distinction from electromagnetism. A static mass generates a static gravitational field, just like a static electrical charge generates a static electrical field. An accelerating charge generates electromagnetic radiation, carried by photons.

However, an accelerating mass generates no gravitational waves, gravitational waves are only generated when the acceleration of the mass is changing. That is when the mass has a non zero “snap” (third order derivative of space”, to the fourth and fifth orders of “Crackle” and “pop”).
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Two Spinor-valued formulations of Einstein-Cartan Equations

The field equations of a Metric theory can be derived from a first order Palatini formalism where an orthonormal frame \theta^a is used and in which the constraint that the dynamical connection is a metric compatible connection Q_{ab}=-\ ^\Gamma\nabla g_{ab}=0 is put in by hand. The field equations read:

\ ^\star F^a{}_{b}\wedge\theta^b = -8\pi{} T^a
\ ^\Gamma\nabla\eta^{ab}=\eta^{abc}\wedge\Theta_c=-8\pi{}\tau^{ab}
where the components of the energy-momentum, T_{ab} and spin tensors, \tau_{abc} are given in terms of the three forms T_{a}=T_{ab}\eta^b, \tau_{ab}=\tau_{abc}\eta^c with \eta^a=\ ^*\theta^a. 
Here \theta^a is an orthonormal “dual” (to tetrad) frame where indices a=(0,1,2,3) are Lorentzian time-space indices of a freely falling frame.

Kinematical differential Multivector framework

As Euler-Lagrange equations of a Lagrangian field theory, these Einstein-Cartan equations lend themselves to many alternative kinematical descriptions. Indeed they may be considered as a conservation law for a certain Sparling 3-form defined on the bundle of orthonormal frames over spacetime, M. They are most elegantly presented as two spinor-valued differential 3-form equations, \ ^\Gamma\nabla(\theta^A{}_{A'}\wedge\theta^{BA'})={\sigma}^{AB}
so that F^A{}_B\wedge\theta^{BA'}={S}^{AA'}, where {\sigma}^{AB} and {S}^{AA'} are known source quantities.
For our purposes the main kinematical objects of a Variational principle are the 1-forms \theta^{AA'}. These are the co-frame, verbein objects that are Einstein’s freely falling elevator.  This dynamical variable, \theta^{AA'}=\theta^{AA'}{}_\mu dx^\mu is a Hermitian matrix-valued one-form from which the (real Lorentzian) metric is given as ds^2=\epsilon_{AB}\epsilon_{A'B'}\theta^{AA'}\otimes\theta^{BB'}.

For Real General Relativity the soldering functor is required to be real (\overline{\theta^a}=\theta^a), \overline{\theta_{\mu}{}^{AA'}}=\theta_{\mu}{}^{AA'}.
In constructing the 4-form Palatini Lagrangian we can make use of higher order multivector constructions based on the co-frame such as \eta^{AA'}=\frac{i}{3}(\theta^{AB'}\wedge\theta^{BA'}\wedge\theta_{BB'})
The internal indices AA' associated to the spin structure over space-time only acquires the interpretation as spinor indices through the dynamical soldering form, \theta^{AA'}{}_\mu. A priori there is no relation between the tangent space and the internal space of the vector bundle B associated to the spinor structure over space time, M. Rather if a (non compact) spacetime manifold, M admits a global null tetrad it has a spinor-structure {PB}defined on it. A spinor structure, PB is a principal fibre bundle with structure group SL(2,\mathbb{C}), the gauge group for spinor dyads. The (real) space-time manifold carries a SL(2,\mathbb{C}) spin (trivial vector) bundle, B associated to PB and its conjugate on it. The tensor product of these two bundles can be identified with the complexified tangent bundle. Each fibre, S\equiv \mathbb{C}^2 of B consists of a 2-complex dimensional vector space equipped with a symplectic metric, \epsilon_{AB}.

Cartan’s Structure Equations

With \Gamma^A{}_B and \bar{\Gamma}^{A'}{}_{B'} (complex conjugate) sl(2,\mathbb{C})-valued connection one-forms and the torsion two form denoted as \Theta^{AA'}, the first Cartan structure equation reads
\Theta^{AA'}:=d\theta^{AA'}-\theta_{AB'}\wedge\bar{\Gamma}^{A'}{}_{B'}-\theta_{BA'}\wedge\Gamma^A{}_B
That is,
\Theta^{AA'}:=\nabla\theta_{AA'}, where \nabla\equiv\ ^\Gamma\nabla denotes the exterior covariant derivative with respect to the sl(2,\mathbb{C})-valued connection(s).
The internal `symplectic metric’, \epsilon_{AB} is given as fixed so that the internal SL(2,\mathbb{C})connection is then traceless \Gamma_{AB}=\Gamma_{BA} due to \nabla\epsilon_{AB}=0.
Defining the basis of anti-self dual two-forms as
\Sigma^{AB}:=\frac{1}{2}\theta^A_{A'}\wedge\theta^{BA'}
The second Cartan structure equations take the complex form
{ F}^A_{\ B}:=d\Gamma^A_{\ B}+\Gamma^A_{C}\wedge\Gamma^C_{\ B}\quad
{ F}^A_{\ B}:={\Psi}^A_{\ BCD}\Sigma^{CD}+{\Phi}^A_{\ BC'D'}\bar\Sigma^{C'D'}+2\Lambda\Sigma^A_{\ B}+(\chi_{D}{}^{A}\Sigma_{B}{}^{D}+\chi_{DB}\Sigma^{AD})
where the curvature two-form, { F}^A_{\ B}, has been decomposed into spinor fields of dimension 5,9,1 and 3 respectively , corresponding to the anti-self dual part of the Weyl conformal spinor, \Psi^A_{\ BCD}, the spinor representation of the trace-free part of the Ricci tensor, -2\Phi^A_{\ BC^\prime D^\prime} and the Ricci scalar 24\Lambda, – all with respect to the curvature of the SL(2,\mathbb{C}) connection and \chi^{AB} arising from the presence of non-zero torsion.
Here we have used the basis of  anti-self dual two-forms \Sigma^{AB}, defined in terms of the co-frame dynamical variable for mere ease of exposition. As we will see in a later post these objects can be treated as the bona fide, fully chiral dynamical “graviton” field object in tis own right.

Semi-Chiral Lagrangian formulation Equations


Two Euler-Lagrange equations arise from Lagrangians of the general form
L_{Tot} = (\Gamma^A{}_B,\theta^{AA'},\rho..), with \theta^{AA'} hermitian and \rho representing some matter fields.
The internal SO(1,3)_{\mathbb{C}}^-\cong SL(2,\mathbb{C}) connection is not associated to the tangent bundle and is thus not a linear connection but a spinor connection. The variation of the Lagrangian with respect to \Gamma^A{}_B will determine this connection in terms of the co-frame so that the bundle B can then be considered soldered to M. The co-frame variation evaluated at the particular value of the connection just determined gives equations for the co-frames only. There exists a unique Levi Civita connection, \omega (with curvature \Omega) so the sl(2,\mathbb{C})connection, \Gamma can be decomposed according to
\Gamma^A{}_B=\omega^A{}_B+K^A{}_B,
F^A{}_B=\Omega^A{}_B+\ ^\omega\nabla K^A{}_B+ K^A{}_C\wedge K_{B}{}^{C}
where K^A{}_B is the contorsion one form, irreducibly written in terms of totally symmetric and`axial’ parts as
K_{AB}=-\frac{1}{2}\sigma_{ABCC'}\theta^{CC'}+2\Theta_{(A|C'|}\theta_{B)}{}^{C'}.
The Einstein-Matter equations owing to the triviality of the Bianchi Identity
^\omega\nabla\Theta^{AA'}=\Omega^A{}_B\wedge\theta^{BA'}+\Omega^{A'}{}_{B'}\wedge\theta^{AB'}=0,
have the simpler form,
^\omega\nabla(\theta^A{}_{A'}\wedge\theta^{BA'})=0,
-2i\Omega^A{}_B\wedge\theta^{BA'}=-8\pi{T}^{AA'}
It is possible to solve for K^A{}_B and replace any \Gamma^A{}_B in { S}^{AA'} by \omega^A{}_B+K^A{}_B.

Semi-Chiral Lagrangian for Fermion fields


The Lagrangian for fermion matter has a dependence on the connection so admits torsion contributions but nevertheless can be written as the sum of a semi-chiral complex Lagrangian for vacuum General Relativity, L_{SC}(\theta,\Gamma), a complex (semi)chiral fermion matter Lagrangian, L_{\frac{1}{2}} and a term, L_{J^2} that ensures the standard Einstein-Weyl form of the field equations,
L_{SC}(\theta,\Gamma)= i\theta^{A}{}_{A'}\wedge\theta^{BA'}\wedge F_{AB},
{L}_{\frac{1}{2}}(\theta,\Gamma,\lambda,\tilde{\lambda})=+\eta^{AA'}\wedge\tilde{\lambda}_{A'}\lambda_{A},
{L}_{J^2}(\lambda,\tilde{\lambda})=\frac{3}{16} \lambda_{A}\tilde{\lambda}_{A'}\lambda^A\tilde{\lambda}^{A'},
L_{Tot}= L_{SC}+ L_{\frac{1}{2}}+ L_{J^2}.
The \lambda_A(\tilde{\lambda}_{A'}) are the left (resp. right)-handed zero forms.
The theory uses only the anti-self dual connection, { D} (which does not act on tensors, so for example
D\theta^{AA'}=d\theta^{AA'}-\theta^{BA'}\wedge\Gamma^A{}_B
but is complete and it turns out, (by varying K^A{}_B), that the real source current, J_{AA'}=\lambda_A\lambda_{A'}=-J_{A'A} supports only the axial part of the torsion of \Gamma^A{}_B,
K_{AB}=-\frac{1}{4}J_{C'(A}\theta_{B)}{}^{C'}.
Because ultimately the real theory is of interest (where \tilde{\lambda}{}_{A'}=\overline{\lambda_A} and \theta is hermitian) it proves useful to extend { D} to \nabla. Although it is argued that the spin \frac{1}{2} field variables can be taken to be either Grassman [or complex]-valued, in fact the use of complex spin \frac{1}{2} fields leads to a non-standard energy-momentum tensor which includes quartic spin \frac{1}{2} fields.
\textrm{Capovilla R, Dell J, Jacobson T and Mason L }\textit{Classical Quantum Gravity,} \textbf{7}, 1990, L1.\textrm{Plebanski, J.F. (1975).} \textit { J. Math. Phys, } \textbf{16}, 2395.\textrm{Plebanski. J.F.} (1974-75).\textit{Spinors,Tetrads and Forms (unpublished)}.\textrm{Robinson DC.} \textit{Classical Quantum Gravity, } \textbf{11}, (1994), L157.\textrm{Robinson DC }\textit {Journal Math. Phys. }, \textbf{36} (7), (1995).
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