8. Forces, Connections and Gauge Fields

1. 8. Forces, Connections and Gauge Fields 8.0. Preliminary 8.1. Electromagnetism 8.2. Non-Abelian Gauge Theories 8.3. Non-Abelian Theories and Electromagnetism 8.4. Relevance of Non-Abelian Theories to Physics 8.5. The Theory of Kaluza and Klein

2. 8.0 Preliminary General relativity: gravitational forces due to geometry of spacetime. Logical steps that lead to this conclusion: 1. Physical quantities (tensors) at different points in spacetime are related by an affine connection, which defines parallel transport. 2. Connection coefficients that cannot be set equal to zero everywhere by a suitable coordinate transformation indicate the presence of gravitational forces. 3. Such effects can be described by a principle of least action. Gravitational forces arises from communication between points in spacetime. Likewise for gauge theories.

3. 8.1. Electromagnetism Internal Space Complex wavefunction: Constant overall phase θ0 is not observable but θ(x) is. E.g. Consider (x) as a vector in the 2-D internal space of the spacetime point x. → Fibre bundle with spacetime as base manifold & internal space the typical fibre. → (x) is a vector field (cross section) of the bundle. → θ(x) gives the orientation of the vector at x.

4. θ0 not observable → parallel transport to define parallelism. Physically significant change is Γ = connection coefficients “Flat” space: Directions of (x) can be referred to one global coordinate system. → (x1) and (x2) are parallel if n = integer → Internal space is the same for all x. → Free particle. “Curved” space : Electromagnetism.

5. Connection Coefficients = (measurable) probability amplitude  ( x1 → x2 ) is physically equivalent to  ( x1 ) → → →  Aμ= electromagnetic vector potential

6. Group Manifold Parallel transport preserves |  | → it affects only phase θ.  Typical fibre is unit circle |  | = 1 or θ [ 0 , 2π). Phase transformation : For θ = const: → e iθ is a symmetry transformation with multiplication is a Lie group called U(1) → The typical fibre θ [ 0 , 2π) is also the (symmetry) group manifold.

7. Local gauge transformation: → gauge tensors on fibre Global gauge transformation:  = Gauge vector * = Gauge 1-form Gauge tensor field of rank (nm) : with

8. Covariant Derivative → Under gauge transformation where Note: D does not change the rank of gauge tensors.

9. Dμ is a gauge vector : → Same as EM gauge transformation  → A μ(x) is called a gauge field. In general, Summary: Phases of a complex wavefunction constitue a U(1) fibre bundle, whose geometry is determined by the gauge fields.

10. Spin ½ Particles • Advantages of geometric point of view of interactions: • Easy generalization. • Provides classification of tensors. E.g., To include the effects of gauge fields, set λ = charge → → Minimal coupling : promotes global to local gauge symmetry In the absence of EM fields, there is a gauge such that everywhere. = 0 → Check: Indeed: 

11. Field Equations → is gauge invariant Simplest scalar under both Lorentz & gauge transformations is with = Maxwell field tensor Action:  Fscales with A, i.e.,  λ ~ coupling strength

12. For system with n types of spin ½ particles : Rescale: Euler-Lagrange equations for A are just the Maxwell equations with (Prove it!) e = elementary charge unit. No restriction of λ derived → charge quantization not explained. Remedy: grand unified theory

13. 8.2. Non-Abelian Gauge Theories 8.2.1. Isospin 8.2.2. Isospin Connection 8.2.3. Field Tensor 8.2.4. Gauge Transformation 8.2.5. Intermediate Vector Boson 8.2.6. Action 8.2.7. Conserved Currents

14. 8.2.1. Isospin Protons and neutrons are interchangeable w.r.t. strong interaction. Conjecture: They are just different states of the nucleon. Nucleon wavefunction : isotopic spin (isospin) state. Proton state: Neutron state: Complete set of independent operators in the isospin space: I, τ Isospin operator = Any unitary operator that leaves * unchanged can be written as θ ~ gauge transformation α ~ rotation in 3-D isospin space Proton and neutron states are the isospin up and down states along z-axis.

15. 8.2.2. Isospin Connection Fibre bundle with spacetime as base manifold & isospin space as typical fibre. Reminder: Directions in isospin space have observable physical meanings. Only meaningful change in isospin space is a rotation. Parallel transport : i, j = p,n 1st order in α: → There is no scale factor because the field tensor does not scale with the gauge fields.

16. Typical fibre can be generated by rotations → SU(2) Gauge covariant derivative : Gauge transformation: → D is a gauge scalar → EM case: U = e i θ(x)  →

17. 8.2.3. Field Tensor  Note: F is nonlinear in A. → F is not gauge invariant & doesn’t scale with A. → Different states of the same isospin must have the same isospin connection. Only particles of different isospins can have different connections.

18. Exact form of F depends on the representation of the gauge group used. Generators of the gauge (Lie) group are T. Corresponding Lie algebra is defined by Cabc = structure constants for SU(2) = εabc

19. 8.2.4. Gauge Transformation By definition, a gauge transformation is a rotation on  given by (  is a gauge vector ) Ta is a generator of the transformation → it is a gauge tensor of rank 2 : → A is not a gauge tensor. = gauge tensor of rank 2 ( proof ! ) →

20. Alternatively, { Ta } is a basis for vector operators on the isospin space. A gauge transformation is then a rotation operator U defined by U b a (α) is determined by comparison with expresses the vector F w.r.t. basis { Ta } Gauge transformation: → or There is an isomorphism between U and U. ~ The SU(2) representation formed by T a is the adjoint representation, so called because

21. 8.2.5. Intermediate Vector Boson Task: Construct a gauge invariant action for the gauge fields. where To ensure that Tr( Fμν Fμν) is a gauge scalar, set → It is straightforward to show that the Pauli matrices satifsy Scaling: Dropping ~ : Quantized gauge fields → intermediate vector bosons (mediate weak interaction) S contains terms like g(A)AA & g2AAAA → IVBs are charged

22. 8.2.6. Action Rescaling by A → gA : where Each jis a 2T(j)+1 multiplet of 4-component Dirac spinors :

23. Euler-Lagrange equations for the field degrees of freedom : or where or For the nucleon doublet : Euler-Lagrange eqautions for the spinor degrees of freedom: (Dirac equations)

24. 8.2.7. Conserved Currents Classical EM: gauge invariance → conservation of charges (μj μ = 0 ). Gauge fields: conservation law is Dμj μ = 0 ( jis covariantly conserved). Note: Dμj μ = 0 does not imply conservation of any physical scalar quantity. Dirac particle: → conservation of charges. For the non-abelian SU(2) gauge group: For the non-Abelian Maxwell equations → is the Noether current associated with the non-Abelian symmetry.  = Fermion + vector bosons flows

25. Components of can be thought of as ‘electric’ and ‘magnetic’ fields Ea and Ba. i.e. → ‘magnetic monopoles’ are allowed Comment: Bai here are not the usual magnetic fields. However, the unified electroweak theories is a non-abelian gauge theory. In that case, genuine magnetic monopoles are allowed.

26. 8.3. Non-Abelian Theories and Electromagnetism Consider with → ~ unification of EM & non-Abelian gauge fields (weak interaction) Technical detail: The U(1) members should be EM gauge transformations so they can’t be eiθI . → Standard representations : →

27. For a general isospin T, Qj = charge of the j-th isospin multiple. In a representation where T 3 is diagonal : Y = hypercharge Gell-Mann- Nishijima relations Largest charge of the multiplets is

28. 8.3.a. Gell-Mann- Nishijima Law The Gell-Mann- Nishijima law was proposed in 1953 to explain the “8-fold way” grouping of “stable” hadrons. “Stable” means no decay if electroweak interactions were absent. Particles ( Q, I, Y ) values Directions of increasing values are Q↗, I3→, and Y↑. Y = S for mesons Y = S + 1 for baryons

29. 8.4. Relevance of Non-Abelian Theories to Physics Pure geometrical consideration of the complex wavefunction → Abelian gauge fields → existence of electromagnetic forces Application to isospin → non-abelian gauge fields (Yang-Mills theories) → nuclear weak interaction Modern version: Fundamental particles are quarks, leptons and quanta of fundamental interactions.

30. 8.5. The Theory of Kaluza and Klein Classical (non-quantum mechanical) theory of Kaluza and Klein unifies gravity and electromagnetism by means of a 5-D spacetime. 5-D spacetime metric tensor A, B 0, 1, 2, 3, 5 with g = metric tensor of the Einstein’s 4-D spacetime. Action for “gravity” : Assumptions: 1. The 5th dimension is space-like, i.e., → 2. gμν and Aμ are independent of x5 and 3. The 5th dimension rolls into a circle of radius r5

31. (a miracle!) with • Objections: • There is no physical justification to the required assumptions. • The theory offers no new observable effects. • Update: • Supergravity and superstring theories also make use of spacetimes of more than 4 dimensions.