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AN INTEGRATION OF GENERAL RELATIVITY AND RELATIVISTIC QUANTUM THEORY Joseph E. Johnson, PhD jjohnson@sc.edu May 5, 2016 Physics Department University of South Carolina I. INTRODUCTION A Linear Vector Space (LVS): 1. 1. 2. 3. Allows two operations: |A>+|B> = |C> and a|A> = |B> A basis |i> for the space gives any vector as |A> = ai |i> A LVS becomes more powerful with two products: A Metric Space (MS) is a LVS with the scalar product: 2. 1. 2. 3. 4. 5. A*B = SAiBi = a number = |A| |B| cos q = <A | B > So we get the ‘metrics of length and angle. Examples: regular space and the unitary scalar product In infinite dimensions this is called a Hilbert space Or in a Riemannian geometry <a|b> = gmn am bn where g is the “metric” A Lie Algebra (LA) is a LVS with the product 1. 1. 2. 3. 2. [Li, Lj] = cijk Lk & the Jacobi Identity Scyc per [[L1, L2],L3]=0 Lie Group : G(t) = etL = 1 +tL + t2L2/2!+… LA Examples: Rotation Sxi2, Translations xi=xi+ai, Lorentz c2t2-r2, Poincare (Lorentz with translations), Unitary SYa*Ya=1, General Linear xi = Gij xj, Markov Sxi = Sxi, Scaling xi=elixi & Heisenberg X, P, I. LA Representations: LA can be represented by matrices Li acting upon vectors |Y> in a metric space: |Y’> = L |Y>. 1. Vectors in a LA representation space are distinguished by a list of internal values that normally represent the eigenvalues of a complete set of commuting operators: |a1, a2, …. > 2. One can also use creation, a+, and annihilation operators, a, that create a given vector from the vacuum: |a1, a2, …. > = a+ a1, a2, …. |0> 3. In this framework, the original LA (Li i = 1, 2….) and the “creation operators” that constitute the representation, a+ a1, a2, …. , both have well defined commutation rules and constitute a larger LA !!! NOTATION 1. We use Greek indices for time & space with values 0, 1, 2, 3 with the metric gmn = (+1, -1, -1, -1), and English indices for space alone. 2. Repeated indices indicate contraction by Einstein convention. 3. Generally X0 = ct, X1 = x, X2 = y … 1. And P0 = E/c, P1 = Px , P2 = Py … FUNDAMENTAL FRAMEWORK 1. The fundamental observables (measurements) in nature L1, L2, … can be represented by an algebraic structure (usually a LA) whose commutators show how these observations (measurements) may interfere with each other. 2. The exponential maps from a LA, M(a) = exp (ai Li), form Lie Groups (LG) where the associated Li are conserved quantities if the transformation group leaves a system invariant. 3. The possible states of nature | > are the vectors in a metric space that form a representation space of the LA of possible observables. 4. That which can exist is a representation space for the algebra of actions (measurements / observations) which can be performed. OPERATORS (OBSERVABLES OR ACTIONS) 1. Operators can generate: 1. 2. 3. 2. Operators can be classified as: 1. 2. 3. Symmetries & associated conserved quantities (momentum) Approximate symmetries (isospin) No associated symmetry but are spectrum generating (position) Global like momentum, angular momentum, … Local like gauge transformations Operators can create LA representations from |0> 1. a+a1,a2,… |0> = |a1,a2,… > FOUR DOMAINS ACCEPTED: 1. RQT: Quantum Theory, Special Relativity, & Field Theory 2. SM: The Standard LA Gauge Model of Particles & Dynamics 3. GR: Einstein’s Theory of General Relativity 4. Inversions: Space, Time, & Particle Conjugation 5. Dark Matter (DM) and Dark Energy (DE) are still not understood and are only peripherally considered. THE PROBLEM: Is, It, Ic spin/stat RQT Well Integrated SM ? ? GR DM DE 1. Quantum theory follows from the Heisenberg algebra: 1. 2. and [E, Time] = iħ I Intrinsic spin follows from the rotation algebra: 1. 3. [Pi, Xj] = -iħ dij I [Si, Sj] = eijk Sk Special Relativity follows from the Lorentz algebra: 1. 2. [Mmn, Mrs] = iħ (gmsMnr + gurMms - gmrMns - gusMmr) and [Mmn, Pl] = iħ (gln Pm - glm Pn) for any vector Pl II. THE EXTENDED POINCARE (EP) ALGEBRA (POINCARE LIE ALGEBRA EXTENDED WITH A FOUR-VECTOR POSITION XN OPERATOR) THE POINCARE ALGEBRA IS IN BLACK. THE EXTENSION WITH XN IS TAN. 6. [I, Pm] = [I, Xn ] = [I, Mmn ] = 0 [Pm, Xn] = iħ gmn I (covarient form of [P , X ] = -iħ d [Pm, Pn] = 0 [Xm, Xn] = 0 [Mmn, Pl] = iħ (gln Pm - glm Pn) [Mmn, Xl] = iħ (gln Xm - glm Xn) 7. [Mmn, Mrs] = iħ (gmsMnr + gurMms - gmrMns - 1. 2. 3. 4. 5. i j ij I Heisenberg LA) gusMmr) THE EP REPRESENTATIONS 1. Define Lmn = Xm Pn – Xn Pm (orbital angular momentum tensor) 1. From which it follows that 2. [Lmn, Pl] = iħ (gln Pm - glm Pn) 3. [Lmn, Xl] = iħ (gln Xm - glm Xn) 4. [Lmn, Lrs] = iħ (gms Lnr + gur Lms - gmr Lns - gus Lmr) One can then define an intrinsic spin four-tensor as 1. Smn = Mmn - Lmn EP DECOMPOSED INTO HEISENBERG (XPI) & HOMOGENEOUS LORENTZ (S) 1. Then the commutators for the intrinsic spin Smn are 2. [Smn, Pl] = 0 3. [Smn, Xl] = 0 4. [Smn, Lrs] = 0 5. [Smn, Srs] = iħ (gms Snr + gur Sms - gmr Sns - gus Smr) 6. Thus the EP algebra becomes the product of the two algebras: EP = XPI * S (Heisenberg * Lorentz) and the full representations are the products of the separate representations. XPI HEISENBERG LA REPRESENTATIONS 1. One can use the position representation 1. Xm | y > = y m | y > or the momentum representation 2. Pm | k > = k m | k > or equivalently diagonalize the mass Pm Pm = m2, the sign of the energy, e( P0), and Pi as 3. |m, e( P0), k > THE LORENTZ GROUP REPRESENTATIONS 1. All representations of the homogeneous Lorentz group have been found by Bergmann and by Gelfand, Neimark, and Shapiro to be given by the two Casimir operators b0 and b1 defined as: 2. b02 + b12 – 1 = ½ gmrgnsSmn Srs where b0 = 0, ½, 1, 3/2, …(|b1|-1) and where b1 is a complex number 3. b0 b1 = - ¼ e mnrs Smn Srs and with the Casimir operator for the rotation subalgebra as 4. S2 which has the spectrum s(s+1) with the total spin s = b0 , b0+1, …, (|b1| - 1) and 5. s = -s, -s+1, ….s-1, s which is the z component of spin. THE STANDARD MODEL (SM) ADMITS ONLY THREE REPRESENTATIONS: 1. b0 = ½, b1= ± 3/2: A Dirac spin s=½ particle where the two signs of b1 are conjugate representations giving four components and the g matrices (quarks & leptons). 2. b0 = 0, b1=1: 3. b0 = 0, b1 = 2: A vector particle with spin s = 0, 1 which can be written as a four-vector Am (like the electromagnetic field potential for photons). (photon, W, Z) A spin s = 0 particle (Higgs) THE EP REPRESENTATION SPACE 1. 2. Thus one had the representation|b0 ,b1 , s, s > Which gives the EP representations as: |k m , b0 , b1 , s, s > = a+k , b0, b1, s, s |0> for the momentum or position representation or 3. | y m , b0 , b1 , s, s > = a+y, b0, b1, s, s |0> for the position representation. III. THE STANDARD MODEL OF PARTICLES & DYNAMICS (SM) 1. A gauge LA model using the unitary groups: U(1) x SU(2) x SU(3) on spin ½ (Dirac) spinors generating the associated 4-vector and scalar bosons responsible for the strong, weak, and electromagnetic forces. 2. This phenomenological (non space-time) algebra contains 1+3+8 = 12 observables (operators) including electrical charge. 1. The fields (particles) that support this group are EP representations (a+….) mixed with EP operators (Pm…). 2. Although the SM is still a work in progress, it has had remarkable success in predicting forms of the interactions, masses, and the dynamical evolution of interacting systems using a Lagrangian for the system. IV. EINSTEIN’S THEORY OF GENERAL RELATIVITY (GR) In GR the metric of space time is curved by the presence of matter and energy as given by the equation 1. Rab - ½ gab R + gabL = (8pG/c4) Tab where Rab is the Ricci tensor, R is its contraction, L is the cosmological constant and Tab is the energy momentum tensor expressed in terms of the particles in the SM theory. 1. The Ricci and Riemann tensor terms on the LHS are defined in terms of the derivatives of the Christoffel symbols which are in turn defined in terms of derivatives of the metric tensor. 2. Terms on the RHS are defined in terms of the energy momentum tensor, Tab, expressed in terms of the operators in the SM theory acting on the state of the system. V. A PROPOSED INTEGRATION THE INTEGRATION ARGUMENT 1. The EP is the Lie algebra of all space time observables with structure constants defined in terms of the Minkowski flat metric giving RQT. 2. As the Minkowski metric is altered by GR, it seems reasonable that this metric, the associated space-time operators, and the defining EP structure constants be generalized to the Riemannian curved space time metric required by GR thus capturing all space-time operators in an Extended Poincare Einstein (EPE) algebra. THE INTEGRATION ARGUMENT 1. This merger with quantum theory can now be done by allowing the metric to be a function of the new four-position operators as gmn -> gmn(X). 2. The energy momentum tensor on the RHS of the Einstein equations cannot come from the RQT but rather from the SM operators acting on the state of the current system. (In practice, it comes from large masses such as black holes or stars and thus the classical expression is valid). 3. Thus GR becomes a bridge between the 15 parameter EP global algebra of space-time observables and the 12 parameter SM gauge algebra of phenomenological observables. THE EPE ALGEBRA 1. Is a ‘kinematical’ foundational algebra that encompasses the global operators of space and time but which does not explain forces or the states of physical systems. 2. The EPE representations must also support the gauge algebras and all interactions. 3. Encompasses the foundations of quantum theory (Heisenberg algebra), special relativity (Poincare algebra), and general relativity (gmn) THE INTEGRATION POSTULATE 1. The gmn in all EP structure constants is now to be taken as a function of the fourposition operators, gmn(X), which in the position representation becomes a function of space-time variables to be determined by the Einstein equations using the energy-momentum tensor density Tmn from SM operators acting on the state of the system. 2. We call this generalized EP LA the Extended Poincare Einstein (EPE) LA RQT – SM – GR INTERDEPENDENCE 1. Thus the “space-time” EPE LA gives the representations (particles) which also must serve as the representation space for the SM “gauge algebra”. 2. The energy momentum tensor density operator defined by the representations in the SM then are to determine by GR the metric that defines the EPE algebra which in turn defines the possible SM representations. THE PROPOSED SOLUTION: Is, It, Ic spin/stat RQT (EPE) gmn (X) L SM Tmn GR new DM particle DE conjectured links in tan DM Dynamics is given by Feynman Paths with the SM Lagrangian DETAILS OF THE PROPOSED INTEGRATION 1 1. The basic commutation rules for EP remain the same for EPE (gmn now= gmn (X) & I has the single eigenvalue “1”.) 2. [Pm, Xn] = iħI gmn(X) so we can now write 3. gmn (X) = (-i/ ħ) [Pm, Xn] 4. In the position representation one now has 5. <y| Pm | Y> = iħ gmn(y) ∂/∂yn) Y(y) 1. = iħ ∂m Y(y) where Y(y) = <y| Y >. DETAILS OF THE PROPOSED INTEGRATION 2 1. Although one still has 1. 2. 3. 4. 2. where from now on 1. 3. [Pm, Pn ] = 0 but since [Pm, gab ] ≠ 0 it follows that [Pm, [Pn, Xr]] ≠ 0 so that the Heisenberg algebra is no longer nilpotent. It also follows that [Pm, gab] = iħ gmn (∂gab /∂yn) = iħ ∂mgab = (-i/ħ )[Pm, [Pa, Xb]] gab = gab (y) is to be understood. Thus in the position representation all ∂m f(x) are converted into (–i/ħ) [Pm, f(X)] DETAILS OF THE PROPOSED INTEGRATION 3 Christoffel symbols are defined as: 1. Gcab = (½) ( [∂b, gca] + [∂ a, gcb] - [∂ c, gab]) 2. Gcab = (½) (-i/ħ) ( [Pb, gca] + [Pa, gcb] - [ Pg, gab]) Then using 3. 4. gab (X) = (-i/ ħ) [Pa, Xb] one obtains Ggab = (-½) (1/ħ2) ( [Pb, [Pg, Xa]] + [Pa, [Pg, Xb]] - [ Pg, [Pa, Xb] ] ) DETAILS OF THE PROPOSED INTEGRATION 4 Then the Riemann tensor becomes: 1. Rlijk = (-i/ħ) ( [Pj, Glik ] - [Pk, Glij ] ) + (Gljs Gsik - Glks Gsij ) 2. Rlijk = (-ħ-2) { ½ ( [Pj, ( [Pk, gli] + [Pi, glk] - [ Pl, gik ] )] - [Pk, ( [Pj, gli] + [Pi, glj] - [ Pl, gij ] )] ) + (( [Ps, glj] + [Pj, gls] - [ Pl, gjs ] ) gsr ( [Pk, gri] + [Pi, grk] - [ Pr, gik ]) - (( [Ps, glk] + [Pk, gls] - [ Pl, gks ] ) gsr ( [Pj, gri] + [Pi, grj] - [ Pr, gij ] )} or in terms of commutators only: DETAILS OF THE PROPOSED INTEGRATION 5 Rlijk = (-i/ħ) (-½) (1/ħ2) ( [Pj, ( [Pk, [Pl, Xi]] + [Pi, [Pl, Xk]] - [ Pl, [Pi, Xk] ] )] - [Pk, ( [Pj, [Pl, Xi]] + [Pi, [Pl, Xj]] - [ Pl, [Pi, Xj] ] ) ] ) + ((-i/ ħ) [Pr, Xs]) (( [Ps, [Pl, Xj]] + [Pj, [Pl, Xs]] - [ Pl, [Pj, Xs] ] ) ( [Pk, [Pr, Xj]] + [Pj, [Pr, Xk]] - [ Pr, [Pj, Xk] ]) - ((-i/ ħ) [Pr, Xs]) ( [Ps, [Pl, Xk]] + [Pk, [Pl, Xs]] - [ Pl, [Pk, Xs] ] ) ( [Pj, [Pr, Xj]] + [Pj, [Pr, Xj]] - [ Pr, [Pj, Xj] ])) DETAILS OF THE PROPOSED INTEGRATION 6 One then defines the Ricci tensor as: Rlj = gik Rlijk = (-i/ ħ) [Pi, Xk] Rlijk and also defines R = glj Rlj or (-i/ ħ) [Pl, Xj] Rlj all to be inserted into the Einstein equations, Rlj - ½ glj R + gljL Rlj + ((i/ ħ) [Pl, Xj]) ( ½ R + L ) = (8 π G/c4) Tlj = (8 π G/c4) Tlj or DETAILS OF THE PROPOSED INTEGRATION 7 1. This expresses the LHS of the GR equations in terms of the fundamental EP operators and their commutators. 2. The energy momentum tensor Tab must now be expressed in terms of the operators in the standard model: 3. Tab = Y (ga Pb)Y where the (ga Pb ) term is both symmetrized and also acts in both directions on all Dirac fields and similar expressions obtain for the gauge bosons. 1. Since the EPE LA describes the structure of all space-time observables, it follows that the generalization of gab to gab(X) incorporates the curved nature of space time generalizing both special relativity and quantum theory. 2. It captures the dynamics of geodesic motion of entities as dictated by the metric. 3. It does not provide information on the allowable particle mass and spin states, nor the interactions of the strong and electroweak force nor the dynamical evolution under those forces all of which is still done in the SM. SYSTEM SOLUTIONS - 1 One might begin with a static environment where particles are in a large gravitational field such as near a black hole or neutron star. Here we can compute the metric as the Schwarzschild solution. Having the metric then we can use the representation of P in the position representation as: <y| Pm | Y> = iħ gmn(y) ∂/∂yn) Y(y) The internal spin tensor is not altered via the metric. Then one can write the representations as before such as for the spin ½ fields for the SM. SYSTEM SOLUTIONS - 2 The Lagrangian can be constructed as before but now with the new representations for Pb and ga . The dynamics would proceed by path-integral solutions and the energy momentum tensor would be computed from the dominant fields (which might just consist of the contribution from the massive sphere or black hole. The particle paths would then follow geodesics in the curved space time in order to minimize the Lagrangian phase interference. VI A: DARK ENERGY (DE) 1. It is currently known that the cosmological constant L in the GR equations is the simplest account for the DE expansion of the universe. 2. And since the accelerated expansion of the universe is a space time transformation, it is natural to assume that DE arises from this GR parameter (until otherwise informed by experiments). VI B: DARK MATTER (DM) 1. It is known that DM is not contained in the current SM as it only interacts gravitationally while the SM represents only the phenomenological LA structure of the strong, electromagnetic, and weak interactions. 2. Thus there is no expectation for it to be an existing known particle. 3. It would be reasonable for DM to be a new particle in the SM like the Higgs, but stable, that naturally would not be predicted from its existing phenomenological basis in other interactions. 4. This assumption then will require an associated Tmn term to account for this unknown particle whose mass and spin would have to be determined from gravitational interactions or theoretical arguments. POSSIBLE TEST AREAS: 1. 2. 3. Study & simplify the 30 parameter algebra of space-time and gauge operations along with the a, a+ representation operations and analyze for patterns. Use the Schwarzschild solution for the metric near a massive object where a test atom, nucleus, or molecule is placed to study possible alterations in the photon emissions. Using the altered uncertainty relations between X and P, seek effects that could be detectable in states of systems. PROPOSED PROGRAM 1. I propose to study the combined 30 parameter algebra composed of the 15 parameter EPE and the 12 parameter gauge LA and adjoining the 3 parameter Inversion algebra. 2. To study this algebra seeking simplifications in the tensor expressions with the Bianchi, Jacobi, and other means with the hope that some ‘beautiful structure’ might emerge as it did with Maxwell’s equations. SPECIFICALLY NEEDED: 1. 2. 3. 4. 5. Formulate the 12 parameter gauge algebra along with the three discrete inversions as algebraic commutators including expressions of all mixing angles and related SM parameters. Study and simplify the EPE and general 30 parameter LA using symbol manipulating software (Mathematica, Python/Sage, MatLab) Study the mathematical properties of the 30 parameter algebra and the associated group manifolds. Seek new predictions involving quantum effects in strong gravitational files Seek new implications in the SM framework THANK YOU THE EPE ALGEBRA (EXTENDED POINCARE EINSTEIN LIE ALGEBRA) [I, Pm] = [I, Xn ] = [I, Mmn ] = 0 [Pm, Xn] = iħ gmn I which is the Heisenberg Lie algebra – the foundation of quantum theory [Pm, Pn] = 0 insuring the noninterference of energy momentum measurements all four dimensions [Xm, Xn] = 0 insuring the noninterference of time and position measurements all four dimensions [Mmn, Pl] = iħ (gln Pm - glm Pn) which guarantees that Pl transforms as a vector under Mmn [Mmn, Xl] = iħ (gln Xm - glm Xn) which guarantees that Xl transforms as a vector under Mmn [Mmn, Mrs] = iħ (gmsMnr + gurMms - gmrMns - gusMmr) which guarantees that Mrs transforms as a tensor TO BEGIN: 1. 2. 3. 4. Assume that any physical system (components of the universe) can be represented by vectors in a linear vector space |a> which is also a metric space: <a|b> = a number. Assume that any action on |a> can be represented by an operator L in linear vector space of fundamental operators, L = aiLi , i = 1, 2, … with closure L|a> = |b> Assume that the Li form a non-commutative algebra of fundamental actions [Li , Lj] = cijk Lk. and which is (normally) a Lie algebra. Thus |a> must be a representation space of this algebra where the order of actions define the interference among the fundamental actions as defined by the structure constants: cijk. THE FUNDAMENTAL ALGEBRAS OF RELATIVITY AND QUANTUM THEORY 1. Quantum theory follows from the Heisenberg algebra: 1. 2. Intrinsic spin follows from the rotation algebra: 1. 3. [Pi, Xj] = -iħ dij I [Si, Sj] = eijk Sk Special Relativity follows from the Lorentz algebra: 1. 2. [Mmn, Mrs] = iħ (gmsMnr + gurMms - gmrMns - gusMmr) and [Mmn, Pl] = iħ (gln Pm - glm Pn) for any vector Pl EXTENSION OF THE POINCARE LIE ALGEBRA TO INCLUDE A POSITION FOUR VECTOR XM When a position operator X1 is viewed by a moving observer, it must transform into a “time operator” X0. Thus we extended the Heisenberg Lie algebra to include a four-vector space-time position operator which obeys [Pm, Xn] = iħ gmu I where “I” commutes with all operators. Thus we are extending the Poincare algebra by five new operators Xn and I forming a 15 parameter Lie algebra of M, P, X, and I. THE EP OR EPE ALGEBRA (EXTENDED POINCARE LIE ALGEBRA) [I, Pm] = [I, Xn ] = [I, Mmn ] = 0 [Pm, Xn] = iħ gmn I which is the Heisenberg Lie algebra – the foundation of quantum theory [Pm, Pn] = 0 insuring the noninterference of energy momentum measurements all four dimensions [Xm, Xn] = 0 insuring the noninterference of time and position measurements all four dimensions [Mmn, Pl] = iħ (gln Pm - glm Pn) which guarantees that Pl transforms as a vector under Mmn [Mmn, Xl] = iħ (gln Xm - glm Xn) which guarantees that Xl transforms as a vector under Mmn [Mmn, Mrs] = iħ (gmsMnr + gurMms - gmrMns - gusMmr) which guarantees that Mrs transforms as a tensor