Unified Field Equations Coupling Four Forces and Theory of Dark Matter and Dark Energy
|
|
- Timothy Cannon
- 6 years ago
- Views:
Transcription
1 Unified Field Equations Coupling Four Forces and Theory of Dark Matter and Dark Energy Tian Ma, Shouhong Wang Supported in part by NSF and ONR fluid 1
2 Outline I. Motivations II. Principle of Interaction Dynamics III. Unified Field Equations Coupling Four Forces IV. Duality V. Electroweak Theory VI. Unified Theory of Dark Energy and Dark Matter VII. Concluding Remarks 2
3 References: 1. Tian Ma & Shouhong Wang, Gravitational Field Equations and Theory of Dark Matter and Dark Energy, June, 2012, 2. Tian Ma & Shouhong Wang, Unified Field Equations Coupling Four Forces and Principle of Interaction Dynamics, September, 2012, 3
4 I. Motivations Mystery of dark matter and dark energy 4
5 In 1998, published observations of Type Ia supernovae by the High-z Supernova Search Team followed in 1999 by the Supernova Cosmology Project suggested that the expansion of the universe is accelerating. Saul Perlmutter, Brian P. Schmidt, Adam G. Riess. Dark energy is a hypothetical form of energy that permeates all of space and tends to accelerate the expansion of the universe. Unfortunately, no theory can explain the nature of dark energy, which is now considered the cosmos greatest mystery. 5
6 Possible explanations: Cosmological constant, which would be a property of the vacuum and would stretch space. Quintessence: Dark energy could be a new type of force field that occupies space. 6
7 Grand challenges for coupling all four interactions 7
8 8
9 9
10 II. Principle of Interaction Dynamics (PID) Let u L 2 (T r s M). We define the operators D A and div A as D A u = Du + u A, div A u = divu A u, where A is a vector or covector field, D and div are the usual gradient and divergent operators. Orthogonal Decomposition Theorem: Let A is a given vector field or covector field, and u L 2 (T r s M) with r + s 1. Then u can be orthogonally decomposed into (1) u = D A ϕ + v with div A v = 0, where ϕ H 1 (Ts r 1 M) or ϕ H 1 (Ts 1M). r 10
11 The classical Einstein equations are derived based on three principles: principle of equivalence and principle of general relativity, which amount to saying that space-time is a Riemanian manifold (M, g ij ), and gravity is described by the metric g ij. Lagrangian least action principle: Einstein-Hilbert functional: L EH = M ( R + 8πG ) gdx c 4 S δl EH = 0 = R ij 1 2 g ijr + 8πG c 4 T ij = 0 11
12 T ij = S ij 1 2 g ijs + g kl S kl g ij, S = gkl S kl, R ij = 1 2 gkl ( 2 g kl x i x j + 2 g ij x k x l + g kl ( g rs Γ r kl Γ s ij Γ r ilγ s kj), Γ k ij = 1 ( gil 2 gkl x j + g jl x i g ) ij x l. 2 g il x j x k 2 g kj x i x l For example, for an ideal fluid case with pressure p, energy density ε, and 4D velocity u µ T µν = (g 00 p + ε)u µ u ν pg uv ) 12
13 Due to the presence of dark energy and dark matter, the energy-momentum tensor T ij of normal matter is no longer conserved: D i (T ij ) 0. By an orthogonal decomposition theorem, there is a scalar function ϕ : M R such that (2) c4 T ij = T ij 8πG D id j ϕ, D i Tij = 0 ( L EH = R + 8πG ) gdx, S M D i [R ij 1 2 g ijr Namely ] c 4 = 0 = R ij 1 2 g ijr + 8πG c 4 T ij = 0 R ij 1 2 g ijr = 8πG c 4 T ij D i D j ϕ, D i ( D i D j ϕ + 8πG ) c 4 T ij = 0 13
14 Equivalently: The new gravitational field equations (2) can be derived with constraint least action principle: lim λ 0 1 λ [L EH(g ij + λx ij ) L EH (g ij )] = (δl EH (g ij ), X) = 0 D i X ij = 0 This constraint least action leads us to postulate: Principle of Interaction Dynamics (PID): For all physical interactions there are Lagrangian actions (3) L(g, A, ψ) = M L(g ij, A, ψ) gdx, where A is a set of vector fields representing the gauge potentials, and ψ are the wave functions of particles. The states (g, A, ψ) are the extremum points of (3) with the (D + A)-free constraint. 14
15 III. Unified Field Equations Coupling Four Forces The unified field equations are derived based on the following principles: the Einstein principle of general relativity (or Lorentz invariance) and the principle of equivalence, which amount to saying that the space-time is a 4-dimensional Riemannian manifold (M, g ij ) the principle of gauge invariance the principle of interaction dynamics (PID) 15
16 Recapitulation of Maxwell Equations: Let (E 1, E 2, E 3 ) = A 0 (A1, A 2, A 3 ), B = curl A t 0 E 1 E 2 E 3 F µν = µ A ν ν A µ, F µν = E 1 0 B 3 B 2 E 2 B 3 0 B 1 E 3 B 2 B 1 0 Lagrangian action: L = 1 4 F µνf µν = 1 2 (E2 B 2 ) Maxwell equation (with no source): µ F µν = 0 16
17 Coupling Dirac theory to Maxwell Quantum Electrodynamics (QED): Dirac equation: (iγ µ µ m)ψ = 0, where ψ is a 4-vector wave function. Here γ µ are Dirac matrices: ( ) ( ) γ 0 I 0 = γ µ 0 σ µ = 0 I σ µ 0 ( ) ( ) ( ) σ = σ 2 0 i = σ = 1 0 i Then the Lagrangian action L = ψ [iγ µ ( µ + iea µ ) m]ψ 1 4 F µνf µν is invariant under the U(1) gauge transformation: ψ(x) e ieλ(x) ψ(x), A µ A µ ν Λ(x). 17
18 Both the field equations below and the Lagrangian action above preserve both the gauge symmetry and the Lorentz symmetry: µ ( µ A ν ν A µ ) eψ γ ν ψ = 0, [iγ µ ( µ + iea µ ) m]ψ = 0 U(1) gauge theory works very well for QED But for the U(1) SU(2) electroweak gauge theory, Glashow-Weinberg-Salam needed to add to the Lagrangian the Higgs terms L H = D µ φ D µ φ λ(φ φ a) 2 + G e ( LφR + Rφ L) for energy creation and mass generation. Here φ = (φ +, φ 0 ) t. Higgs mechanism: The added Higgs terms spontaneously breaks the gauge symmetry in the field equations, but not in the action. 18
19 Field Equations Coupling Four Forces Lagrangian action functional L is the sum of the Einstein-Hilbert L EH, the GWS L GW S without Higgs field terms, and the standard QCD L QCD. (4) L = [L EH + L GW S + L QCD ] gdx L EH = R + 8πG c 4 S, L GW S = 1 4 W a µνw aµν 1 4 F µνf µν + k { Lk (iγ µ Dµ m k )L k + ψ } k R (iγ µ Dµ m l k)ψk R, L QCD = 1 4 F b µνf bµν + 6 k=1 q k (iγ µ Dµ m q k )q k 19
20 (5) W a µν = D µ W a ν D ν W a µ + g 1 f abc W b µw c ν, F µν = D µ A ν D ν A µ, F b µν = D µ B b ν D ν B b µ + g 2 g bcd B c µb d ν, D µ L k = ( µ i g 1 2 W a µσ a + i e 2 A µ)l k, D µ ψ R k = ( µ + iea µ )ψ R k, D µ q k = ( µ + i g 2 2 Bb µλ b )q k, where g 1 and g 2 are constants, f abc and g bcd are the structure constants of SU(2) and SU(3), σ a are the Pauli matrices, λ b are the Gell-Mann matrices, L k = (ψ νk, ψk L)t are the wave functions of left-hand lepton and quark pairs (each has 3 generations), ψk R are the right-hand leptons and quarks, q k = (q k1, q k2, q k3 ) t is the k-th flavored quark, and µ is the Lorentz Vierbein covariant derivative. 20
21 Thanks to the div A -free constraint, the Euler-Lagrangian of the action functional is balanced by gradient fields, resulting the unified field equations (6) coupling all four forces. The gradient fields break spontaneously the gauge-symmetries. This gives rise to a complete new mechanism for spontaneous gauge symmetry-breaking and for energy and mass generation, which provides similar outcomes as the Higgs mechanism. 21
22 Unified Field Equations Coupling Four Forces: (6) R µν 1 2 g µνr + 8πG c 4 T µν = (D µ + α 0 A µ + α W a Wµ a + α Bj Bµ)Φ j ν, 6 [ ] 1 µ ( µ A ν ν A µ ) e 2 L k γ ν L k + ψ k R γ ν ψk R k=1 = (D ν + β 0 A ν + β W a W a ν + β Bj B j ν)φ 0, µ W a µν g b=1 f bca g µα W a µνw c α + g k=1 L k γ ν σ a L k = (D ν + κ a 0A ν + κ a W bw b ν + κ a BjB j ν)φ a W, a = 1, 2, 3, µ F b µν g k=1 g klb g µα B b µνb l α g k=1 q k γ ν λ b q k = (D ν + θ b 0A ν + θ b W lw l ν + θ b BlB l ν)φ b B, b = 1,, 8, (iγ µ Dµ m k )L k = 0 (iγ µ Dµ m l k)ψ R k = 0 22 (iγ µ Dµ m q k )q k = 0.
23 T µν = δs + c4 [ δg ij 16πG gαβ WαµW a βν a + F αµ F βν + FαµF b βν b ] c 4 16πG g µν(l GW S +L QCD ). 23
24 By the Bianachi identity, taking divergence on the field equations, we have D µ D µ Φ + (divv 1 )Φ = o(1), D µ D µ φ 0 + (divv 2 )φ 0 = o(1), D µ D µ φ a W + (divv a 3 )φ a W = o(1) for a = 1, 2, 3, D µ D µ φ b B + (divv b 3 )φ b B = o(1) for b = 1,, 8, which imply that Φ represents a vector boson, and φ 0, φ a W, φb B bosons. Here represent scalar V 1 = α 0 A µ + α W a W a µ + α Bj B j µ, V 2 = β 0 A ν + β W a W a ν + β Bj B j ν, V a 3 = κ a 0A ν + κ a W bw b ν + κ a BjB j ν, V b 3 = θ j 0 A ν + θ j W b W b ν + θ j Bl Bl ν. 24
25 IV. Duality The unified field equations (6) coupling all four forces induce a natural duality: (7) {g µν } Φ µ, A µ φ 0, Wµ a φ a W for a = 1, 2, 3, Bµ b φ b B for b = 1,, 8. Equivalently the graviton g, the photon γ, vector bosons W ± and Z, and the gluons g b, represented by the left-hand side of the above duality corresponds to the bosonic fields: a vector massless spin-1 boson Φ µ, a scalar massless spin-0 boson φ 0, and scalar massive spin-0 bosons φ a W, φb B. 25
26 Zero-point energy: The field φ 0, adjoint to the electromagnetic potential A µ, is a massless field with spin s = 0, and it needs to be confirmed experimentally. We think it is this particle field φ 0 that causes the vacuum fluctuation or zero-point energy. Eightfold Way mesons: We conjecture that the combination of the eight scalar bosons φ b B corresponding to gluons g b (1 b 8) leads to the Eightfold Way mesons: π +, π, π 0, K +, K, K 0, K0, η 0. 26
27 Higgs Bosons: The combination of the three Higgs bosonic fields on the right hand side of the duality induces three Higgs boson particles given by φ ± = 1 2 (φ 1 W ± iφ 2 W ), φ 3 W with φ ± having ± electric charges, and with φ 3 W being neutral. Note that the classical Weinberg-Salam theory induces one neutral Higgs boson particle. All three Higgs bosons deduced here possess masses, generated by the new mechanism. In fact, consider only the electromagnetic and weak interactions and ignore the effect of other interactions, we derive a totally different electroweak theory. Again this electroweak theory produces the three vector bosons W ± and Z, as well as the three Higgs bosons. The spontaneous gauge-symmetry breaking is achieved by the constraint action without Higgs terms in the action functional. 27
28 V. Electroweak Theory Glashow-Weinberg-Salam electroweak theory: The action density is given by L = L G + L F + L H, L G = 1 4 W µνw a aµν 1 4 F µνf µν, L F = i Rγ µ D µ R + i Lγ µ D µ L, L H = D µ φ D µ φ λ(φ φ a) 2 + G e ( LφR + Rφ L) Here φ = ( φ +, φ 0) t is the scalar field. Three components of this φ induce masses for bosonic force careers W ± and Z, and the remaining one defines the neutral Higgs boson. 28
29 New electroweak theory based on PID: The Lagrangian action (8) L = [ 1 4 W a µνw aµν 1 4 F µνf µν + L(iγ µ D µ m)l + R(iγ ] µ D µ m e )R dx, where Wµν k = µ Wν k ν Wµ k + λf kij WµW i ν j, F µν = µ A ν ν A µ, D µ = µ + iea µ + iλwµσ k k, D µ = µ + iea µ. Note: When no weak interaction is present, W k µν = 0, ψ L ν = 0, and the action (8) reduces to the classical QED action. 29
30 SU(2) gauge transformation: L e (i/2)θ k σ k L, φ e (i/2)θ k σ k φ, R R, W k µ W k µ 2 g 1 µ θ k + f kij θ i W j µ, B µ B µ U(1) gauge transformation: L e (i/2)β L, φ e (i/2)β φ, R e iβ R, Wµ k Wµ k 2 µ β, g 2 B µ B µ + 2 µ β. g 2 30
31 Electroweak interacting field equations (based on PID): (9) (10) (11) (12) µ W k µν λ 2 3 f lik WµνW k µ i + Jν k, l=1 = ( ν + α k Wν k + α 0 A ν )φ k 1 for k = 1, 2, 3, µ F µν + Jν L + Jν R = ( ν + β j Wν j + β 0 A ν )φ 2, (iγ µ D µ m)l = 0, (iγ µ D µ M e )R = 0, where J k ν = λ Lγ ν σ k L J L = e Lγ ν L, J R = e Rγ ν R. weak neutral current 31
32 Mass generation for W ± and Z: Let the scalar function φ k 1 be in the form φ 1 = ρ k + φ k, ρ k = constant, φ k 0. Then omitting higher order terms, equations (9) become (13) µ W k µν m k W k ν = 0, m k = ρ k α k, for k = 1, 2, 3. Let m 1 = m 2 = M W, m 3 = M Z. Then we derive three vector bosons with respective masses as follows: W ± µ = 1 2 (W 1 µ ± iw 2 µ), Z µ = W 3 µ, which are the same as those derived from the Weinberg-Salam theory. 32
33 Higgs bosons: Taking divergence on both sides of (9) and noticing that ν µ Wνµ k = 0, we obtain that (14) µ µ φ k 1 = m k φ k 1 + higher order terms, where m k is the constant term in α k µ W k µ + α 0 µ A µ = m k + higher order terms. Hence (14) can be regarded as the Higgs field equations, and φ k 1 are the Higgs particles with mass m k. Weak neutral current: J k ν. 33
34 VI. Unified Theory of Dark Energy and Dark Matter The new vector particle field Φ µ is massless with spin s = 1. The interaction between this particle field Φ µ and the graviton leads to a unified theory of dark matter and dark energy and explains the acceleration of expanding universe. 34
35 Consider the gravitational field equations alone: R µν 1 2 g µνr = 8πG ( ) 8πG c 4 T µν D µ D ν ϕ, D µ c 4 T µν + D µ D ν ϕ R = 8πG c 4 T + Φ, Φ gdx = 0 The new spin-1 bosonic massless particle field is now given by the Φ µ = D µ ϕ, satisfying the Klein-Gorden equation in the vacuum (T µν = 0): M D ν D ν (D µ ϕ) = 0. = 0 If the matter in the universe is uniformly distributed, we can show that Φ µ = D µ ϕ = 0. Hence the new particle field will have no effect. This energy density c4 8πGΦ consists of both positive and negative energies. The negative part produces attraction, and the positive part produces repelling force. 35
36 Consider a spherically symmetric central matter field with mass M and radius r 0. In this case, (15) ds 2 = g 00 c 2 dt 2 + g 11 dr 2 + r 2 (dθ 2 + sin 2 θdϕ 2 ), and physically g 00 is given by (16) g 00 = (1 + 2c ) 2ψ, where ψ is the Newton gravitational potential. The tensors g ij in (15) are written as (17) g 00 = e u, g 11 = e v, g 22 = r 2, g 33 = r 2 sin 2 θ, u = u(r), v = v(r). 36
37 The field equations become: (18) (19) (20) u + 2u r + u 2 (u v ) = ϕ 1 2 (u + v 4 r )ϕ, u 2v r + u 2 (u v ) = ϕ (u + v + 4 r )ϕ, u v + 2 r (1 ev ) = r(ϕ (u v )ϕ ). 37
38 With the new field equations, we derive that the force exerted on an object with mass m is given by (21) F = mmg [ 1r 2 c2 2GM ( 2 + 2GM ) 1 dϕ c 2 r dr + ] c2 2GM Φ r for r > r 0. The first term is the classical Newton gravitation, the second term is the coupling interaction between matter and the scalar potential ϕ, and the third term is the interaction generated by the scalar potential energy density Φ (R = Φ for r > r 0 ). 38
39 The sum ε = ε 1 + ε 2 of the scalar potential energy density ε 1 = c4 8πG Φ and the coupling interaction energy between the matter field and the scalar potential field ϕ ( ε 2 = c4 2 8πG r + 2MG ) dϕ c 2 r 2 dr, unifies dark matter and dark energy: ε > 0 ε < 0 represents dark energy, represents dark matter. 39
40 The interaction force formula (21) can be further simplified to derive the following approximate formula for r 0 < r < r km: (22) F = mmg [ 1 r 2 k ] 0 r + k 1r, where k 0 = km 1 and k 1 = km 3, which are estimated using rotation curves of galactic motion. Again, in (22), the first term represents the Newton gravitation, the attracting second term stands for dark matter and the repelling third term is the dark energy. 40
41 VII. Concluding Remarks Principle of Interaction Dynamics: a general principle for coupling interactions. Mechanism of energy creation and mass generation, completely different from the Higgs mechanism. Unified field equations coupling all fours forces, leading to results agreeable with those produced by the standard model and hence experimentally tested. Duality between the force carriers and the new bosonic particle fields. New predictions such as the charged Higgs bosons. The interaction of the new massless, spin-1 bosonic particle Φ µ, and the graviton offers a unified theory of dark matter and dark energy. 41
Unified Field Equations Coupling Force Forces. Tian Ma, Shouhong Wang Supported in part by NSF and ONR
Unified Field Equations Coupling Force Forces Tian Ma, Shouhong Wang Supported in part by NSF and ONR http://www.indiana.edu/ fluid 1 Outline I. Motivations II. PID III. Unified Field Equations Coupling
More informationTwo Fundamental Principles of Nature s Interactions
Two Fundamental Principles of Nature s Interactions Tian Ma, Shouhong Wang Supported in part by NSF, ONR and Chinese NSF http://www.indiana.edu/ fluid I. Gravity and Principle of Interaction Dynamics PID)
More informationLaw of Gravity and Gravitational Radiation
Law of Gravity and Gravitational Radiation Tian Ma, Shouhong Wang Supported in part by NSF and ONR http://www.indiana.edu/ fluid Blog: https://physicalprinciples.wordpress.com I. Laws of Gravity, Dark
More informationPhysical Laws of Nature vs Fundamental First Principles
Physical Laws of Nature vs Fundamental First Principles Tian Ma, Shouhong Wang Supported in part by NSF, ONR and Chinese NSF http://www.indiana.edu/ fluid I. Laws of Gravity, Dark Matter and Dark Energy
More informationLaw of Gravity, Dark Matter and Dark Energy
Law of Gravity, Dark Matter and Dark Energy Tian Ma, Shouhong Wang Supported in part by NSF, ONR and Chinese NSF Blog: http://www.indiana.edu/~fluid https://physicalprinciples.wordpress.com I. Laws of
More informationUnified Theory of Dark Energy and Dark Matter
Unified Theory of Dark Energy and Dark Matter Tian Ma, 1 Shouhong Wang 2 1 Department of Mathematics, Sichuan University, Chengdu, P. R. China 2 Department of Mathematics, Indiana University, Bloomington,
More informationNew Blackhole Theorem and its Applications to Cosmology and Astrophysics
New Blackhole Theorem and its Applications to Cosmology and Astrophysics I. New Blackhole Theorem II. Structure of the Universe III. New Law of Gravity IV. PID-Cosmological Model Tian Ma, Shouhong Wang
More informationUNIFIED FIELD EQUATIONS COUPLING FOUR FORCES AND PRINCIPLE OF INTERACTION DYNAMICS
UNIFIED FIELD EQUATIONS COUPLING FOUR FORCES AND PRINCIPLE OF INTERACTION DYNAICS TIAN A AND SHOUHONG WANG Abstract. The main objective of this article is to postulate a principle of interaction dynamics
More informationUNIFIED FIELD THEORY AND PRINCIPLE OF REPRESENTATION INVARIANCE
UNIFIED FIELD THEORY AND PRINCIPLE OF REPRESENTATION INVARIANCE TIAN MA AND SHOUHONG WANG Abstract. This is part of a research program to establish a unified field model for interactions in nature. The
More informationAula/Lecture 18 Non-Abelian Gauge theories The Higgs Mechanism The Standard Model: Part I
Física de Partículas Aula/Lecture 18 Non-Abelian Gauge theories The The : Part I Jorge C. Romão Instituto Superior Técnico, Departamento de Física & CFTP A. Rovisco Pais 1, 1049-001 Lisboa, Portugal 2016
More informationThe Gauge Principle Contents Quantum Electrodynamics SU(N) Gauge Theory Global Gauge Transformations Local Gauge Transformations Dynamics of Field Ten
Lecture 4 QCD as a Gauge Theory Adnan Bashir, IFM, UMSNH, Mexico August 2013 Hermosillo Sonora The Gauge Principle Contents Quantum Electrodynamics SU(N) Gauge Theory Global Gauge Transformations Local
More informationLecture 8. September 21, General plan for construction of Standard Model theory. Choice of gauge symmetries for the Standard Model
Lecture 8 September 21, 2017 Today General plan for construction of Standard Model theory Properties of SU(n) transformations (review) Choice of gauge symmetries for the Standard Model Use of Lagrangian
More informationYang-Mills Gravity and Accelerated Cosmic Expansion* (Based on a Model with Generalized Gauge Symmetry)
review research Yang-Mills Gravity and Accelerated Cosmic Expansion* (Based on a Model with Generalized Gauge Symmetry) Jong-Ping Hsu Physics Department, Univ. of Massachusetts Dartmouth, North Dartmouth,
More informationParticle Physics I Lecture Exam Question Sheet
Particle Physics I Lecture Exam Question Sheet Five out of these 16 questions will be given to you at the beginning of the exam. (1) (a) Which are the different fundamental interactions that exist in Nature?
More informationSymmetry Groups conservation law quantum numbers Gauge symmetries local bosons mediate the interaction Group Abelian Product of Groups simple
Symmetry Groups Symmetry plays an essential role in particle theory. If a theory is invariant under transformations by a symmetry group one obtains a conservation law and quantum numbers. For example,
More informationarxiv: v2 [physics.gen-ph] 11 Jul 2012
arxiv:1206.5078v2 [physics.gen-ph] 11 Jul 2012 GRAVITATIONAL FIELD EQUATIONS AND THEORY OF DARK ATTER AND DARK ENERGY TIAN A AND SHOUHONG WANG Abstract. The main objective of this article is to derive
More informationGenesis of Electroweak. Unification
Unification Tom Kibble Imperial College London ICTP October 2014 1 Outline Development of the electroweak theory, which incorporates the idea of the Higgs boson as I saw it from my standpoint in Imperial
More informationBack to Gauge Symmetry. The Standard Model of Par0cle Physics
Back to Gauge Symmetry The Standard Model of Par0cle Physics Laws of physics are phase invariant. Probability: P = ψ ( r,t) 2 = ψ * ( r,t)ψ ( r,t) Unitary scalar transformation: U( r,t) = e iaf ( r,t)
More informationHunting New Physics in the Higgs Sector
HS Hunting New Physics in the Higgs Sector SM Higgs Sector - Test of the Higgs Mechanism Oleg Kaikov KIT, Seminar WS 2015/16 Prof. Dr. M. Margarete Mühlleitner, Dr. Roger Wolf, Dr. Hendrik Mantler Advisor:
More informationQED and the Standard Model Autumn 2014
QED and the Standard Model Autumn 2014 Joel Goldstein University of Bristol Joel.Goldstein@bristol.ac.uk These lectures are designed to give an introduction to the gauge theories of the standard model
More information9 Quantum Field Theory for Children
101 9 Quantum Field Theory for Children The theories (known and hypothetical) needed to describe the (very) early universe are quantum field theories (QFT). The fundamental entities of these theories are
More informationElectroweak Theory & Neutrino Scattering
Electroweak Theory & 01.12.2005 Electroweak Theory & Contents Glashow-Weinberg-Salam-Model Electroweak Theory & Contents Glashow-Weinberg-Salam-Model Electroweak Theory & Contents Glashow-Weinberg-Salam-Model
More informationGauge Symmetry in QED
Gauge Symmetry in QED The Lagrangian density for the free e.m. field is L em = 1 4 F µνf µν where F µν is the field strength tensor F µν = µ A ν ν A µ = Thus L em = 1 (E B ) 0 E x E y E z E x 0 B z B y
More informationNon Abelian Higgs Mechanism
Non Abelian Higgs Mechanism When a local rather than global symmetry is spontaneously broken, we do not get a massless Goldstone boson. Instead, the gauge field of the broken symmetry becomes massive,
More informationAn Introduction to the Standard Model of Particle Physics
An Introduction to the Standard Model of Particle Physics W. N. COTTINGHAM and D. A. GREENWOOD Ж CAMBRIDGE UNIVERSITY PRESS Contents Preface. page xiii Notation xv 1 The particle physicist's view of Nature
More information752 Final. April 16, Fadeev Popov Ghosts and Non-Abelian Gauge Fields. Tim Wendler BYU Physics and Astronomy. The standard model Lagrangian
752 Final April 16, 2010 Tim Wendler BYU Physics and Astronomy Fadeev Popov Ghosts and Non-Abelian Gauge Fields The standard model Lagrangian L SM = L Y M + L W D + L Y u + L H The rst term, the Yang Mills
More informationIntroduction to gauge theory
Introduction to gauge theory 2008 High energy lecture 1 장상현 연세대학교 September 24, 2008 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, 2008 1 / 72 Table of Contents 1 Introduction 2 Dirac equation
More informationCurrent knowledge tells us that matter is made of fundamental particle called fermions,
Chapter 1 Particle Physics 1.1 Fundamental Particles Current knowledge tells us that matter is made of fundamental particle called fermions, which are spin 1 particles. Our world is composed of two kinds
More informationPARTICLE PHYSICS Major Option
PATICE PHYSICS Major Option Michaelmas Term 00 ichard Batley Handout No 8 QED Maxwell s equations are invariant under the gauge transformation A A A χ where A ( φ, A) and χ χ ( t, x) is the 4-vector potential
More informationOption 212: UNIT 2 Elementary Particles
Department of Physics and Astronomy Option 212: UNIT 2 Elementary Particles SCHEDULE 26-Jan-15 13.00pm LRB Intro lecture 28-Jan-15 12.00pm LRB Problem solving (2-Feb-15 10.00am E Problem Workshop) 4-Feb-15
More informationA model of the basic interactions between elementary particles is defined by the following three ingredients:
I. THE STANDARD MODEL A model of the basic interactions between elementary particles is defined by the following three ingredients:. The symmetries of the Lagrangian; 2. The representations of fermions
More informationElementary particles and typical scales in high energy physics
Elementary particles and typical scales in high energy physics George Jorjadze Free University of Tbilisi Zielona Gora - 23.01.2017 GJ Elementary particles and typical scales in HEP Lecture 1 1/18 Contents
More informationField Theory and Standard Model
Field Theory and Standard Model W. HOLLIK CERN SCHOOL OF PHYSICS BAUTZEN, 15 26 JUNE 2009 p.1 Why Quantum Field Theory? (i) Fields: space time aspects field = quantity φ( x, t), defined for all points
More informationTHE INSTITUTE FOR SCIENTIFIC COMPUTING AND APPLIED MATHEMATICS
Field Theory for Multi-Particle System Tian Ma and Shouhong Wang July 16, 2014 Preprint No.: 1404 THE INSTITUTE FOR SCIENTIFIC COMPUTING AND APPLIED MATHEMATICS INDIANA UNIVERSITY FIELD THEORY FOR MULTI-PARTICLE
More informationBirth of electroweak theory from an Imperial perspective
Birth of electroweak theory from an Imperial perspective Tom Kibble King s College London 2 Oct 2012 Electroweak theory Oct 2012 1 Outline Story of spontaneous symmetry breaking in gauge theories and electro-weak
More informationGeometry and Physics. Amer Iqbal. March 4, 2010
March 4, 2010 Many uses of Mathematics in Physics The language of the physical world is mathematics. Quantitative understanding of the world around us requires the precise language of mathematics. Symmetries
More informationOn the W ± and Z 0 Masses
On the W ± and Z Masses Kenneth Dalton e-mail: kxdalton@yahoo.com Abstract Scalar and vector fields are coupled in a gauge invariant manner, such as to form massive vector fields. In this, there is no
More informationIntroduction to particle physics Lecture 6
Introduction to particle physics Lecture 6 Frank Krauss IPPP Durham U Durham, Epiphany term 2009 Outline 1 Fermi s theory, once more 2 From effective to full theory: Weak gauge bosons 3 Massive gauge bosons:
More informationarxiv:quant-ph/ v5 1 Aug 2005
Constraints of the Dynamic Equations and Lagrangian required by Superposition of Field X. Sun a, Z. Yang b,c a Institute of High Energy Physics, Beijing 100039, China a Graduate University of the Chinese
More informationWeek 3: Renormalizable lagrangians and the Standard model lagrangian 1 Reading material from the books
Week 3: Renormalizable lagrangians and the Standard model lagrangian 1 Reading material from the books Burgess-Moore, Chapter Weiberg, Chapter 5 Donoghue, Golowich, Holstein Chapter 1, 1 Free field Lagrangians
More informationThe θ term. In particle physics and condensed matter physics. Anna Hallin. 601:SSP, Rutgers Anna Hallin The θ term 601:SSP, Rutgers / 18
The θ term In particle physics and condensed matter physics Anna Hallin 601:SSP, Rutgers 2017 Anna Hallin The θ term 601:SSP, Rutgers 2017 1 / 18 1 Preliminaries 2 The θ term in general 3 The θ term in
More informationTHE STANDARD MODEL AND THE GENERALIZED COVARIANT DERIVATIVE
THE STANDAD MODEL AND THE GENEALIZED COVAIANT DEIVATIVE arxiv:hep-ph/9907480v Jul 999 M. Chaves and H. Morales Escuela de Física, Universidad de Costa ica San José, Costa ica E-mails: mchaves@cariari.ucr.ac.cr,
More informationVertex Based Missing Mass Calculator for 3-prong Hadronically Decaying Tau Leptons in the ATLAS Detector
Vertex Based Missing Mass Calculator for 3-prong Hadronically Decaying Tau Leptons in the ATLAS Detector Harvey Maddocks MPhys Physics Department of Physics Lancaster University September 2014 A thesis
More informationNew Fundamental Wave Equation on Curved Space-Time and its Cosmological Applications
New Fundamental Wave Equation on Curved Space-Time and its Cosmological Applications Z.E. Musielak, J.L. Fry and T. Chang Department of Physics University of Texas at Arlington Flat Space-Time with Minkowski
More informationPotential Descending Principle, Dynamic Law of Physical Motion and Statistical Theory of Heat
Potential Descending Principle, Dynamic Law of Physical Motion and Statistical Theory of Heat Tian Ma and Shouhong Wang Supported in part by NSF, ONR and Chinese NSF http://www.indiana.edu/ fluid Outline
More informationChapter 2 The Higgs Boson in the Standard Model of Particle Physics
Chapter The Higgs Boson in the Standard Model of Particle Physics.1 The Principle of Gauge Symmetries The principle of gauge symmetries can be motivated by the Lagrangian density of the free Dirac field,
More informationNovember 24, Scalar Dark Matter from Grand Unified Theories. T. Daniel Brennan. Standard Model. Dark Matter. GUTs. Babu- Mohapatra Model
Scalar from November 24, 2014 1 2 3 4 5 What is the? Gauge theory that explains strong weak, and electromagnetic forces SU(3) C SU(2) W U(1) Y Each generation (3) has 2 quark flavors (each comes in one
More informationFundamental Particles and Forces
Fundamental Particles and Forces A Look at the Standard Model and Interesting Theories André Gras PHYS 3305 SMU 1 Overview Introduction to Fundamental Particles and Forces Brief History of Discovery The
More informationREVIEW. Quantum electrodynamics (QED) Quantum electrodynamics is a theory of photons interacting with the electrons and positrons of a Dirac field:
Quantum electrodynamics (QED) based on S-58 Quantum electrodynamics is a theory of photons interacting with the electrons and positrons of a Dirac field: Noether current of the lagrangian for a free Dirac
More informationLecture 16 V2. October 24, 2017
Lecture 16 V2 October 24, 2017 Recap: gamma matrices Recap: pion decay properties Unifying the weak and electromagnetic interactions Ø Recap: QED Lagrangian for U Q (1) gauge symmetry Ø Introduction of
More informationLecture 4 - Relativistic wave equations. Relativistic wave equations must satisfy several general postulates. These are;
Lecture 4 - Relativistic wave equations Postulates Relativistic wave equations must satisfy several general postulates. These are;. The equation is developed for a field amplitude function, ψ 2. The normal
More informationProblem Set #4: 4.1, 4.3, 4.5 (Due Monday Nov. 18th) f = m i a (4.1) f = m g Φ (4.2) a = Φ. (4.4)
Chapter 4 Gravitation Problem Set #4: 4.1, 4.3, 4.5 (Due Monday Nov. 18th) 4.1 Equivalence Principle The Newton s second law states that f = m i a (4.1) where m i is the inertial mass. The Newton s law
More informationLattice Gauge Theory and the Maxwell-Klein-Gordon equations
Lattice Gauge Theory and the Maxwell-Klein-Gordon equations Tore G. Halvorsen Centre of Mathematics for Applications, UiO 19. February 2008 Abstract In this talk I will present a discretization of the
More informationLectures on Chiral Perturbation Theory
Lectures on Chiral Perturbation Theory I. Foundations II. Lattice Applications III. Baryons IV. Convergence Brian Tiburzi RIKEN BNL Research Center Chiral Perturbation Theory I. Foundations Low-energy
More informationA brief introduction to modified theories of gravity
(Vinc)Enzo Vitagliano CENTRA, Lisboa May, 14th 2015 IV Amazonian Workshop on Black Holes and Analogue Models of Gravity Belém do Pará The General Theory of Relativity dynamics of the Universe behavior
More informationPhysics 129 LECTURE 6 January 23, Particle Physics Symmetries (Perkins Chapter 3)
Physics 129 LECTURE 6 January 23, 2014 Particle Physics Symmetries (Perkins Chapter 3) n Lagrangian Deductions n Rotations n Parity n Charge Conjugation Gauge Invariance and Charge Conservation The Higgs
More informationConformal Symmetry Breaking in Einstein-Cartan Gravity Coupled to the Electroweak Theory
Conformal Symmetry Breaking in Einstein-Cartan Gravity Coupled to the Electroweak Theory J. Lee Montag, Ph.D. December 018 Abstract We develop an alternative to the Higgs mechanism for spontaneously breaking
More informationElementary Particles II
Elementary Particles II S Higgs: A Very Short Introduction Higgs Field, Higgs Boson, Production, Decays First Observation 1 Reminder - I Extend Abelian Higgs model to non-abelian gauge symmetry: ( x) +
More informationQuantum Field Theory 2 nd Edition
Quantum Field Theory 2 nd Edition FRANZ MANDL and GRAHAM SHAW School of Physics & Astromony, The University of Manchester, Manchester, UK WILEY A John Wiley and Sons, Ltd., Publication Contents Preface
More informationQuantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams
Quantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams III. Quantization of constrained systems and Maxwell s theory 1. The
More informationParticle Physics. experimental insight. Paula Eerola Division of High Energy Physics 2005 Spring Semester Based on lectures by O. Smirnova spring 2002
experimental insight e + e - W + W - µνqq Paula Eerola Division of High Energy Physics 2005 Spring Semester Based on lectures by O. Smirnova spring 2002 Lund University I. Basic concepts Particle physics
More informationGeneral Relativity and Differential
Lecture Series on... General Relativity and Differential Geometry CHAD A. MIDDLETON Department of Physics Rhodes College November 1, 2005 OUTLINE Distance in 3D Euclidean Space Distance in 4D Minkowski
More informationAnisotropic Interior Solutions in Hořava Gravity and Einstein-Æther Theory
Anisotropic Interior Solutions in and Einstein-Æther Theory CENTRA, Instituto Superior Técnico based on DV and S. Carloni, arxiv:1706.06608 [gr-qc] Gravity and Cosmology 2018 Yukawa Institute for Theoretical
More informationGeneral Relativity Lecture 20
General Relativity Lecture 20 1 General relativity General relativity is the classical (not quantum mechanical) theory of gravitation. As the gravitational interaction is a result of the structure of space-time,
More informationGravitational radiation
Lecture 28: Gravitational radiation Gravitational radiation Reading: Ohanian and Ruffini, Gravitation and Spacetime, 2nd ed., Ch. 5. Gravitational equations in empty space The linearized field equations
More informationQuantum Electrodynamics and the Higgs Mechanism
Quantum Electrodynamics and the Higgs Mechanism Jakob Jark Jørgensen 4. januar 009 QED and the Higgs Mechanism INDHOLD Indhold 1 Introduction Quantum Electrodynamics 3.1 Obtaining a Gauge Theory..........................
More informationAspects of Spontaneous Lorentz Violation
Aspects of Spontaneous Lorentz Violation Robert Bluhm Colby College IUCSS School on CPT & Lorentz Violating SME, Indiana University, June 2012 Outline: I. Review & Motivations II. Spontaneous Lorentz Violation
More informationPRINCIPLES OF PHYSICS. \Hp. Ni Jun TSINGHUA. Physics. From Quantum Field Theory. to Classical Mechanics. World Scientific. Vol.2. Report and Review in
LONDON BEIJING HONG TSINGHUA Report and Review in Physics Vol2 PRINCIPLES OF PHYSICS From Quantum Field Theory to Classical Mechanics Ni Jun Tsinghua University, China NEW JERSEY \Hp SINGAPORE World Scientific
More informationGauge Invariance. 1.1 Introduction
1 1 Gauge Invariance 1.1 Introduction Gauge field theories have revolutionized our understanding of elementary particle interactions during the second half of the twentieth century. There is now in place
More informationIntercollegiate post-graduate course in High Energy Physics. Paper 1: The Standard Model
Brunel University Queen Mary, University of London Royal Holloway, University of London University College London Intercollegiate post-graduate course in High Energy Physics Paper 1: The Standard Model
More informationGeneration of magnetic fields in the early universe through charged vector bosons condensate
Generation of magnetic fields in the early universe through charged vector bosons condensate JCAP 1008:031,2010 Essential Cosmology for the Next Generation 2011 A. Dolgov, A. Lepidi, G. P. Centro Tecnologico,
More information3.3 Lagrangian and symmetries for a spin- 1 2 field
3.3 Lagrangian and symmetries for a spin- 1 2 field The Lagrangian for the free spin- 1 2 field is The corresponding Hamiltonian density is L = ψ(i/ µ m)ψ. (3.31) H = ψ( γ p + m)ψ. (3.32) The Lagrangian
More informationAs usual, these notes are intended for use by class participants only, and are not for circulation. Week 8: Lectures 15, 16
As usual, these notes are intended for use by class participants only, and are not for circulation. Week 8: Lectures 15, 16 Masses for Vectors: the Higgs mechanism April 6, 2012 The momentum-space propagator
More informationParticles and Strings Probing the Structure of Matter and Space-Time
Particles and Strings Probing the Structure of Matter and Space-Time University Hamburg DPG-Jahrestagung, Berlin, March 2005 2 Physics in the 20 th century Quantum Theory (QT) Planck, Bohr, Heisenberg,...
More informationThe hadronization into the octet of pseudoscalar mesons in terms of SU(N) gauge invariant Lagrangian
The hadronization into the octet of pseudoscalar mesons in terms of SU(N gauge invariant Lagrangian National Research Nuclear University Moscow 115409, Moscow, Russia E-mail: a kosh@internets.ru By breaking
More informationThe Standard Model of Electroweak Physics. Christopher T. Hill Head of Theoretical Physics Fermilab
The Standard Model of Electroweak Physics Christopher T. Hill Head of Theoretical Physics Fermilab Lecture I: Incarnations of Symmetry Noether s Theorem is as important to us now as the Pythagorean Theorem
More informationLectures on the Symmetries and Interactions of Particle Physics
Lectures on the Symmetries and Interactions of Particle Physics James D. Wells CERN Theory, Geneva, Switzerland, and Physics Department, University of Michigan, Ann Arbor, MI USA July 4-10, 013, CERN Summer
More informationGeneral Relativity and Cosmology Mock exam
Physikalisches Institut Mock Exam Universität Bonn 29. June 2011 Theoretische Physik SS 2011 General Relativity and Cosmology Mock exam Priv. Doz. Dr. S. Förste Exercise 1: Overview Give short answers
More informationThe Higgs Mechanism and the Higgs Particle
The Higgs Mechanism and the Higgs Particle Heavy-Ion Seminar... or the Anderson-Higgs-Brout-Englert-Guralnik-Hagen-Kibble Mechanism Philip W. Anderson Peter W. Higgs Tom W. B. Gerald Carl R. François Robert
More informationEspansione a grandi N per la gravità e 'softening' ultravioletto
Espansione a grandi N per la gravità e 'softening' ultravioletto Fabrizio Canfora CECS Valdivia, Cile Departimento di fisica E.R. Caianiello NFN, gruppo V, CG Salerno http://www.sa.infn.it/cqg , Outline
More informationThe Strong Interaction and LHC phenomenology
The Strong Interaction and LHC phenomenology Juan Rojo STFC Rutherford Fellow University of Oxford Theoretical Physics Graduate School course Lecture 2: The QCD Lagrangian, Symmetries and Feynman Rules
More informationPhysics 4213/5213 Lecture 1
August 28, 2002 1 INTRODUCTION 1 Introduction Physics 4213/5213 Lecture 1 There are four known forces: gravity, electricity and magnetism (E&M), the weak force, and the strong force. Each is responsible
More informationQuantum Field Theory. and the Standard Model. !H Cambridge UNIVERSITY PRESS MATTHEW D. SCHWARTZ. Harvard University
Quantum Field Theory and the Standard Model MATTHEW D. Harvard University SCHWARTZ!H Cambridge UNIVERSITY PRESS t Contents v Preface page xv Part I Field theory 1 1 Microscopic theory of radiation 3 1.1
More informationWeak interactions and vector bosons
Weak interactions and vector bosons What do we know now about weak interactions? Theory of weak interactions Fermi's theory of weak interactions V-A theory Current - current theory, current algebra W and
More informationUNIFICATION OF GRAVITATIONAL AND STRONG NUCLEAR FIELDS. Alpha Institute for Advanced Study Received 19 October 2003
UNIFICATION OF GRAVITATIONAL AND STRONG NUCLEAR FIELDS M. W. Evans Alpha Institute for Advanced Study E-mail: emyrone@aol.com Received 19 October 2003 Using the recently derived Evans wave equation of
More informationQFT Dimensional Analysis
QFT Dimensional Analysis In the h = c = 1 units, all quantities are measured in units of energy to some power. For example m = p µ = E +1 while x µ = E 1 where m stands for the dimensionality of the mass
More informationGauge coupling unification without leptoquarks Mikhail Shaposhnikov
Gauge coupling unification without leptoquarks Mikhail Shaposhnikov March 9, 2017 Work with Georgios Karananas, 1703.02964 Heidelberg, March 9, 2017 p. 1 Outline Motivation Gauge coupling unification without
More informationAnomaly. Kenichi KONISHI University of Pisa. College de France, 14 February 2006
Anomaly Kenichi KONISHI University of Pisa College de France, 14 February 2006 Abstract Symmetry and quantization U A (1) anomaly and π 0 decay Origin of anomalies Chiral and nonabelian anomaly Anomally
More informationLecture 19. November 2, Comments on HW3. Introduction to EWK symmetry breaking. Hydro Model of a Quark-Gluon Plasma Tianyu Dai
Lecture 19 November 2, 2017 Comments on HW3 Introduction to EWK symmetry breaking Hydro Model of a Quark-Gluon Plasma Tianyu Dai 1 Class Plans Date Lecture Essay presentation (15 minutes) Nov. 2 19 Tianyu
More informationStephen Blaha, Ph.D. M PubHsMtw
Quantum Big Bang Cosmology: Complex Space-time General Relativity, Quantum Coordinates,"Dodecahedral Universe, Inflation, and New Spin 0, 1 / 2,1 & 2 Tachyons & Imagyons Stephen Blaha, Ph.D. M PubHsMtw
More informationSearch for direct production of sleptons in proton-proton collisions at. the ATLAS detector. s = 13 TeV with. Even Simonsen Håland
Search for direct production of sleptons in proton-proton collisions at s = 13 TeV with the ATLAS detector Even Simonsen Håland Thesis submitted for the degree of Master in Subatomic Physics 60 credits
More informationGauge Theory of Gravitation: Electro-Gravity Mixing
Gauge Theory of Gravitation: Electro-Gravity Mixing E. Sánchez-Sastre 1,2, V. Aldaya 1,3 1 Instituto de Astrofisica de Andalucía, Granada, Spain 2 Email: sastre@iaa.es, es-sastre@hotmail.com 3 Email: valdaya@iaa.es
More informationThis means that n or p form a doublet under isospin transformation. Isospin invariance simply means that. [T i, H s ] = 0
1 QCD 1.1 Quark Model 1. Isospin symmetry In early studies of nuclear reactions, it was found that, to a good approximation, nuclear force is independent of the electromagnetic charge carried by the nucleons
More informationSupersymmetry, Dark Matter, and Neutrinos
Supersymmetry, Dark Matter, and Neutrinos The Standard Model and Supersymmetry Dark Matter Neutrino Physics and Astrophysics The Physics of Supersymmetry Gauge Theories Gauge symmetry requires existence
More informationMSci EXAMINATION. Date: XX th May, Time: 14:30-17:00
MSci EXAMINATION PHY-415 (MSci 4242 Relativistic Waves and Quantum Fields Time Allowed: 2 hours 30 minutes Date: XX th May, 2010 Time: 14:30-17:00 Instructions: Answer THREE QUESTIONS only. Each question
More informationJoint Undergraduate Lecture Tour Higgs Physics and the Mystery of Mass. Heather Logan
1 CAP-CASCA Joint Undergraduate Lecture Tour 2009 Heather Logan With thanks to St. Mary s U., Acadia U., St. Francis Xavier U., Mount Allison U., & U. de Moncton 2 The Large Hadron Collider (LHC) is a
More informationPAPER 45 THE STANDARD MODEL
MATHEMATICAL TRIPOS Part III Friday, 6 June, 014 1:0 pm to 4:0 pm PAPER 45 THE STANDARD MODEL Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
More informationParticle Notes. Ryan D. Reece
Particle Notes Ryan D. Reece July 9, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation that
More informationCHAPTER II: The QCD Lagrangian
CHAPTER II: The QCD Lagrangian.. Preparation: Gauge invariance for QED - 8 - Ã µ UA µ U i µ U U e U A µ i.5 e U µ U U Consider electrons represented by Dirac field ψx. Gauge transformation: Gauge field
More information11 Group Theory and Standard Model
Physics 129b Lecture 18 Caltech, 03/06/18 11 Group Theory and Standard Model 11.2 Gauge Symmetry Electromagnetic field Before we present the standard model, we need to explain what a gauge symmetry is.
More information