Fundamentals of Relativistic Quantum Theory & Integration of ...
On Integrating General Relativity with Quantum Theory by Reframing Riemannian Geometry As a Generalized Heisenberg Lie Algebra MARCH 1, 2018 JOSEPH E. JOHNSON, PHD D E PA R T M E N T O F P H Y S I C S A N D A S T R O N O M Y UNIVERSITY OF SOUTH CAROLINA FEBRUARY 6, 2018 02/06/2020 VERSION DRAFT 1 I. Introduction The Problem 1.
Introduction - The Problem 2-6 2. The Integration Assumptions 7-10 3. Einsteins Equations as a Lie Algebra 11-14 4. Conceptual Foundations
5. Standard HA and PA Representations 6. The SM Gauge Algebra Overview 7. Conclusions & Initiatives 8. Riemannian Geometry as a GLA 9. Thank you
02/06/2020 15-19 20-26 27-32 33-36 37-45 46 2 There are three highly accurate fundamental systems in Physics: 1. Quantum Theory (QT) (including special relativity & field theory) 1.
Based upon the Heisenberg (HA) and Poincare (PA) Lie algebras of space time observables 2. Standard Model (SM) (phenomenological model of particles & interactions) 1. Based upon the Yang Mills SU(3) x SU(2) x U(1) gauge Lie algebras of observables 3. General Relativity (GR) (accelerated systems and gravitation) 1. 02/06/2020 Based upon Einsteins nonlinear diff. eqs. for the Riemann metric of a curved space-time. 3 But Their Union is Deeply Problematic
1. QT and the SM are tightly integrated as Lie algebras of observables a happy marriage 1. 2. 2. But GR describing the gravitational force lives in a totally different world: 1. 2. 3. 4. 5. 3. The strong, electromagnetic, and weak forces are described by the SM as particle exchanges Particles are the representations of these Lie algebras via creation (a +) & annihilation (a) operators.
The observables are not even operators but are space time events x m and the Riemann metric gmu(x) Thus there is no Lie algebra of operators corresponding to the observables in GR and thus There are no Lie algebra representations for quantized particles Instead there are nonlinear differential equations for the curved metric g mu (x) of space-time The gravitational force is described as a geodesic in the space-time defined by g mu (x). GR would appear to be mathematically incompatible with QT and the SM ! a. 02/06/2020 This has been a fundamental problem for 100 years. 4 The Basics of QT & SM: QT has two basic sets of observables: 1. Space-time operators: Xm = (ct, x, y, z) and 2. Energy-momentum operators Pm = (E/c, Px, Py, Pz)
3. They form the Heisenberg Lie algebra (HA) [Pm , Xn ] = i gmn Where gmn = (+1, -1, -1, -1) on the diagonal (a flat space time) 4. Along with the homogeneous Lorentz algebra (LA) gives the Poincare algebra (PA) with Lmn = Xm Pn - Xn Pm Also adjoined are the inversions of space, time, and particle conjugation. The SM describes other observables like charge, isospin, 1. These internal observables form another (gauge) Lie algebra: 2. SU(3) x SU(2) x U(1) that give interactions and particle properties. 02/06/2020 5 Consider the Foundation of GR: GR is based upon two fundamental observables: 1. Space-time events: variables (ct, x, y, z) = xm where m = 0, 1, 2, 3 2. The Riemann curved space-time metric gmn (xl) that determines
the invariant distance between two events. The metric is determined by Einstein's differential equations. 02/06/2020 6 II. The Integration Assumptions 02/06/2020 7 An Integrated Theory (IT) Should Satisfy: 1. When G is negligible (small masses and distances), 1. The IT must give QT & SM - exactly.
2. When is negligible (large masses & large spatial scales), 1. The IT must give GR exactly. It also should be as simple as possible 02/06/2020 8 Recall the HA in Quantum Theory [E, t] = i & [Pm, Xn] = i I gmu [P, X] = -i thus
E = i / t & Px = -i /x where gmu = gmu = (+1, -1, -1, -1) the diagonal Minkowsky metric Where Pm = (E/c, Px, Py, Pz) and Xn = (ct, x, y, z) The metric gives mass: m2c2 = gmu Pm Pn & invariant length: dt2 = gmu dXm dXn The position diagonal representation gives: Xm | y > = ym| y > and = k m | k > where |k> = |k0, k1, k2, k3> Or the mass and the sign of the energy and momenta: Pm Pm = m2 , e( P0), with eigenstates |m, e( P0), k >
02/06/2020 9 Consider the Basic Lie Algebra of QT: 1) The HA is [Pm , Xm ] = i gmn . 1) This is the foundation of QT. 2) It already contains the foundational operators for GR. 3) Position operators Xm for space time (via their eigenvalues). 4)
The metric gmn but for a flat (constant) space-time. 2) Thus with one assumption, the observables of GR (X m and gmn ) can be merged with the observables of QM. 1) Let the HA be generalized so that gmn position representation). = gmn(Xm) where gmn(Xm) is the solution to Einsteins equations (in the 3) This generalizes the concept of a Lie algebra as the structure constants are no longer constants. 02/06/2020 10 III. Einsteins Equations as a
Lie algebra 02/06/2020 11 Conversion of Einsteins Equations to Algebraic (commutator) Form: (rename P as D ) m m 1. [Xm, Xn] = 0 as always but now allow g to be a function of X: 2. [Dm, Xn] = i I gmu(X) so we can now also write 3. gmu (X) = (-i/ ) [Dm, Xn]
Rlabg = (-i/)( [Db, Glag ] - [Dg, Glab ] )+(Glbs Gsag - Glgs Gsab ) which gives the Ricci tensor as Rab = gmn Rambn = (-i/ ) [Dm, Xn] Rambn and the contracted Ricci tensor as R = gab Rab = (-i/ ) [Da, Xb] Rab all of which are inserted into Einsteins equations Rab - gab R + gabL = (8 G/c4) Tab giving the final form as
Rab + ((i/ ) [Pa, Xb]) ( R - L ) = (8 G/c4) Tab are the Einstein equations, where the energy momentum tensor is Tab = as determined by the SM. 02/06/2020 13 These commutation rules define the integration of QM and GR into an algebra 1. But this Lie algebra has structure constants that are dependent upon position in space-time 2. However, in a small local region where the metric can be solved (e.g. the Schwarzschild or Kerr solutions), one can find representations of a standard Lie algebra. 3. Note that the commutation rules produce exactly the equations of standard GR but with operators.
02/06/2020 14 IV. Conceptual Foundations 02/06/2020 15 The Conceptual Foundation of QT: There are actions or measurements that can be performed: These are best represented by operators forming a (Lie) algebra The commutators inform us of how those actions interfere. Matrices or differential operators with a Lie product [Li, Lj] = cijk Lk For example with rotations: [Jx, Jy] = i Jz
The representations of the Lie algebra then Constitute physical systems upon which actions can be executed And thus constitute the particles which can exist in reality. Vector spaces or Hilbert spaces with an inner (scalar) product: Y*(y) Y(y) dy 02/06/2020 16 Fundamental Observables: (Actions and measurements that can be taken on a physical system) Eg. position, momentum, ang. mom., mass, energy, charge, hypercharge Observables are represented by operators that form a Linear Vector Space (LVS) L = ai Li where Li is a complete basis in the (n dimensional) LVS. If [A,B] 0 then the associated observations interfere knowledge is limited Because the order of the observations give different results
This interference is the foundation of quantum theory and the uncertainty principle. If the LVS supports a product with closure in n dimensions then [Li , Lj] = cijk Lk The numbers cijk, called the structure constants, totally define the algebra If antisymmetric and with Jacobi identity then it is a Lie algebra 02/06/2020 17 Fundamental States of a Physical System (upon which actions are taken) The fundamental things (quanta / particles) are LA representations |a1, a2, ..am> = a+ a1, a2, ..am |0> Representations expressed by a+ a1, a2, ..am allow creation/annihilation of states (quanta)
1. gmu (X) = (-i/ ) [Dm, Xn] 2. This means that the metric is ALSO determined by [Dm, Xn] 3. Specifically: the presence of energy/momentum defines the interference of observables. 4.
20 Heisenberg Lie Algebra: This Lie algebra is the foundation of quantum theory 1. The HA: [Pm, Xu] = i gmu where gmu is the flat Minkowsky metric. 2. From it one gets the Heisenberg uncertainty relations X P /2 3. With position diagonal: = -i /y 4. The Fourier transform follows from: 1. can be found from by letting P act in both directions: 2.
= -i /y = k which can be solved as =exp(i/ ) ky (the Fourier transform) 3. These are the directional cosines between the |y> and |k> basis vectors. 02/06/2020 21 The Poincare algebra (Mmn, Pm) is the global symmetry algebra for physics 1. The four momentum Pm generates translations. 2. The Lorentz algebra is generated by the six Mmn = - Mnm 1. Mij generate the rotation group (x2 + y2 + z2 is invariant) 2. M0j generate the Lorentz transformations (c2t2 - x2 - y2 - z2 is invariant)
1. Defined by the following structure constants: 1. [Mmn, Mrs] = i (gmsMnr + gnrMms - gmrMns - gnsMmr) and 2. [Mmn, Pl] = i (gln Pm - glm Pn) 3. [Pm, Pn] = 0 02/06/2020 22 I Extended the Poincare algebra to include the Heisenberg algebra (to a 15 observable algebra): Defined by [Pm, Xn] = i I gmu [Mmn, Xl] = i (gln Xm - glm Xn) [Xm, Xn] = 0 [I, Pm] = [I, Xn ] = [I, Mmn ] = 0 Where I has the eigenvalue 1
02/06/2020 23 The EP representations can be now split into two Lie algebras, Heisenberg and Lorentz Define the orbital angular momentum tensor Lmn = Xm Pn Xn Pm Define the spin tensor Smn = Mmn - Lmn then [Smn, Pl] = 0 ; [Smn, Xl] = 0; [Smn, Lrs] = 0 [Smn, Srs] = i (gms Snr + gur Sms - gmr Sns - gus Smr) The six Smn constitute the homogeneous Lorentz algebra The total angular momentum is Mmn = Lmn + Smn Smn commute with the Pm, Xm , (the Heisenberg algebra) Thus EP = Lorentz + Heisenberg 02/06/2020 24
The homogeneous Lorentz Algebra Representations Two Casimir operators b0 and b1 defined as: b02 + b12 1 = gmr gns Smn Srs b0 b1 = - e mnrs Smn Srs where b0 = 0, , 1, 3/2, (|b1|-1) and where b1 is a complex number S2 = s(s+1) as the rotation Casimir operator as with the total spin s = b0 , b0+1, , (|b1| - 1) s = -s, -s+1, .s-1, s 02/06/2020 = the total spin and
= the z component of spin 25 The EP representations are then the product of the two representations |k m , b0 , b1 , s, s > = a+ k ,b , b , s,s |0> 0 1 |y m , b0 , b1 , s, s > = a+ y , b0,b1,s,s |0> These representations give all possible particles but also give particles we do not see: all masses and all spins. 02/06/2020
26 VI. The SM Gauge Algebra Overview 02/06/2020 27 02/06/2020 28 Abelian Gauge Transformations in EM Interactions: Principle of Minimal EM Interaction The Dirac equation for a charged particle is (gmDm - m) |Y>=0 where Dm = Pm Am(x) It is invariant under the position dependent gauge
transformation |Y> -> eiL(X) |Y> Iff Am(x) -> Am(x) - m L(x) where Am is the electromagnetic four-vector potential This was the precursor to the Yang Mills Non-Abelian gauge transformations. 02/06/2020 29 Yang Mills Non-Abelian Gauge Transformations in the SM Under the Lie Group U(1) x SU(2) x SU(3). The SM requires invariance for |Y> -> eiL(X) |Y> with L(x) transforming under U(1) x SU(2) x U(3) This follows if
30 Require U(1) x SU(2) x SU(3) gauge invariance of the SM by changing Pm -> Dm= Pm Am(X) Recall the commutation relations for the EP algebra allow for
In the quantum chromodynamics (QCD) sector the SU(3) symmetry is generated by Ta while Gam is the SU(3) gauge field containing the gluons. These act only on the quark fields with the strong coupling constant gs. In the electroweak sector Bm is the U(1) gauge field and Yw is the weak hypercharge generating the U(1) group while Wm is the three component SU(2) gauge field and tL are the Pauli matrices which generate the SU(2) group (where the subscript L indicates that they only act on left fermions). The coupling constants are g and g. 02/06/2020 32 VII. Conclusions and Initiatives 02/06/2020
33 The Schwarzschild Metric for a Massive Sphere One of the few solutions of Einsteins equations is for the metric a distance r from a sphere of radius rs and mass M: g00 = 1 rs /r g11 = -1/(1 rs /r) where rs = 2GM/c2 [Dm, Xn] = i I gmu gives the usual uncertainty relationship X0 D0 = ct E/c = t E /2 and Xi Di /2But now we get: 02/06/2020 34
A New Uncertainty Relationship With the Schwarzschild metric one gets: X0 D0 = (t) (E) (/2)(1 rs /r) and X1 D1 (/2)(1/(1 rs /r)) so that their products are unchanged t E x px (/2)2 A more general expression of uncertainty. 02/06/2020 35 The Generalized Fourier Transform alters the wave-particle duality: 1.
The Fourier transform is now more complex: 1. = ( i gmu(y) (/yn) ) = km 2. Which basically is a differential equation for a function whose derivative is proportional to itself 3. 4. Previously with a flat gmu we got = (2p)-2 exp (gmu km yn) But now the metric is a function of space time and we cannot solve this differential equation generally. 2. However in a small region of space-time, near a large massive body (star), we know the Schwarzschild solution for gmu and it can be considered constant over that region thus allowing one to solve the
equation with 1. = (2p)-2 exp (gmu(y) km yn) where gmu(y) is the Schwarzschild constant values. 2. There are now three effective values for gmu(y) km yn : the m = 0 and the m = 1 value (toward the star) from the Schwarzschild solution while the m = 2 and the m = 3 values are as before. This would alter the solutions for a particle in a box. 3. 02/06/2020 36 Next Steps, Complexities Solve the Dirac eq. for hydrogen in a strong gravitational field
Are the altered energy transitions different from the normal theory and observable? Study the uncertainty principle and particle in a box in a Schwarzschild domain Are there observable effects Quantize the gravitational field as bo =0, b1 = 3, spin 2 Poincare symmetric tensor representation. Attempt to formulate a covariant expression of the associated gauge transformations with m=0 Interpret Parallel transport & Geodesics with generating groups of motion Is this theory renormalizable with the graviton? 02/06/2020 37 VI. Riemannian Geometry Formulated as a Generalized Lie Algebra
02/06/2020 38 An Abelian Lie Algebra Eigenvector Space 1. Consider an Abelian Lie algebra of n independent Hermitian linear operators [Xm, Xn] = 0 m, n = 0, 1, 2, 3, 4, 5, . (n-1) Note that this admits higher dimensional space-time manifolds 2. Consider a Hilbert space of square integrable complex functions |Y> as a representation space 1. Let the scalar product normalize the representation space vectors to unity: Y*YY = =1. 3. The representation space has eigenvalues ym for real eigenvectors | ym > 1. Thus Xm | y > = ym | y > where | y > = | y0> >| y1 >| y2 >| | yn-1 > 4. These independent variables can be thought of as the coordinates of an ndimensional space Rn
02/06/2020 39 An Alternate Set of Coordinates 1. Now consider n independent functions of these operators: a. Xm = Xm (X) b. Thus [Xm, Xn] = 0 2. A representation space with eigenvalues y m for real eigenvectors | y m > is a. Xm | y > = ym | y > where | y > = | y0> >| y1 >| y2 >| | yn-1 > 3. Thus ym = ym (ym) defines a transformation from coordinates ym to ym 02/06/2020
40 Contravariant and Covarient Vectors 1. It now follows that dym = ( ym/ yn) dyn and any set of n functions Vm(y) that transform as the coordinates, Vm(y) = ( ym/ yn) Vn(y) is called a contravariant vector 2. The derivatives / yn can be shown to transform as /y'm = ( yn/ ym) / yn and any such vector Vm(y) which transforms in this manner as Vm(y) = ( yn/ ym) Vn(y) is defined as a covariant vector 3. Upper indices are defined as contravariant indices while lower indices are covariant indices.
4. Functions with multiple upper and lower indices that transform as the contravariant and covariant indices just shown are defined as tensors of the rank of the associated indices. 02/06/2020 41 Translation Generators in the Rn Space 1. One would like to have transformations that move one around in this space. 2. It is sufficient to have transformations, Dm, that move one infinitesimally, respectively, in
each direction ym by a given amount as one can generate any finite translation using the group generated by the elements of the Lie algebra via the exponential map: G(h) =exp(h mDm) using the group transformation Xl = G X l G-1. 3. By taking hm to be infinitesimal, then to first order one gets X l = exp(hmDm) X l exp(-hnDn) = (1 +hmDm) X l (1 - hnDn) = X l + hm [ Dm, X l] + higher order terms in h. 4. Thus the commutator [Dm, X l] defines the way in which the transformations interact with each other in executing the transformation in keeping with the theory of Lie algebras and Lie groups. 02/06/2020 42
The Resulting Generalized Lie Algebra 1. We extend the previous n parameter Abelian algebra with an operator I that commutes with the entire basis of the algebra and with a new set of n operators D m that are dual to the operators Xm and which are defined by the Lie brackets [Dm , Xn ] = I gmn(X), with 2. [Dm , I ] = 0 = [Xm , I], and [Xm , Xn] = 0 and where gmn(X) is a function of the Xn operators yet to be defined. 3. This results in a 2n+1 dimensional generalized Heisenberg Lie algebra with D m , Xm , and I as basis operators where I has the unique eigenvalue i where is a fixed real number. 02/06/2020
43 1. This new operator, Dm , can be realized on the basis vectors of the representation space where Xn is diagonal as: 2.
Note that [Dm, Xm] = I gmn(X) means that this is a generalization of the normal definition of a Lie algebra since the metric is now a function of the position operators X which, in the position representation |y>, become the eigenvalues which determine the position in the n dimensional space. 5. Consequently, this Lie Algebra has structure constants, gmn(y), which vary from point to point in the space 02/06/2020 44 1. Since [Dm, Xn] = I gmu(X) one can write gmu (X) = (-i/) [Dm, Xn] implying that the metric tensor gmu(X) is determined by the [Dm, Xn] commutator bracket and that it also is the metric for the
space spanned by the eigenvalues. 2. So in the position representation (basis) one has the results presented above for the [D m, Dn] commutator and the Christoffel symbols and the Ricci and Riemann tensors giving a complete foundation for Riemannian geometry with all derivatives replaced by commutators with Dm and the metric replaced by (-i/) [Dm, Xn]. 02/06/2020 45 Conclusions on RG as a GLA 1. Now the Christoffel symbols, and the Riemann and Ricci tensors are obtained as before. 2.
Does Lie algebra theory impose constraints on gmu(X) 3. Specifically on the topology of space time. 4. Note that the number of spatial dimensions are not restricted to 3 but could contain other hidden dimensions as with string theories. 02/06/2020 46 Thank You for this Opportunity Integration Paper: http://arxiv.org/abs/1606.00701 & www.asg.sc.edu Riemannian Geometry Paper: www.asg.sc.edu
This Slide Show: www.asg.sc.edu Email: [email protected] Web: www.asg.sc.edu Office: Room 405 Physical Science Center University of South Carolina, Columbia SC, 29208 02/06/2020 47 02/06/2020
48 The Heisenberg algebra is given by: [E, t] = i and [P, X] = -i It can be represented as operators acting on functions: E = i / t Px = -i /x Then inserted into E = P2/2m + V(x) gives i/t Y(x) = ((-2/2m)2/x2 + V(x) )Y(x) 02/06/2020 49
A relativistic covariant form is Space-time operators: Xm = (ct, x, y, z) Energy-momentum operators Pm = (E/c, Px, Py, Pz) They form the Covariant Heisenberg Lie algebra (HA) [Pm , Xm ] = i gmn Where gmn = (+1, -1, -1, -1) on the diagonal (a flat space time) 02/06/2020 50
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