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 θ 0 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. θ 0 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. →  (x 1 ) and  (x 2 ) are parallel if n = integer → Internal space is the same for all x. → Free particle. “Curved” space : Electromagnetism.

5. Connection Coefficients = (measurable) probability amplitude  ( x 1 → x 2 ) is physically equivalent to  ( x 1 ) → →→  A μ = 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 : → e iθ is a symmetry transformation is a Lie group called U(1) For θ = const: with multiplication → The typical fibre θ  [ 0, 2π) is also the (symmetry) group manifold.

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

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

9. D μ is a gauge vector : →  In general, Same as EM gauge transformation → A μ (x) is called a gauge field. 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 : Proton state:Neutron state: isotopic spin (isospin) 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 C abc = structure constants for SU(2) = ε abc

19. 8.2.4.Gauge Transformation By definition, a gauge transformation is a rotation on  given by (  is a gauge vector ) T a 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, { T a } 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 { T a } 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( F μν F μν ) 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 & g 2 AAAA → IVBs are charged

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

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 (  μ j μ = 0 ). Gauge fields: conservation law is D μ j μ = 0 ( j  is covariantly conserved). Note: D μ j 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 E a and B a. i.e. → ‘magnetic monopoles’ are allowed Comment: B ai 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 Considerwith → ~ 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 e iθ I. → Standard representations : →

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

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. ( Q, I, Y ) values Particles Directions of increasing values are Q ↗, I 3 →, 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 5 th dimension is space-like, i.e., 2. g μν and A μ are independent of x 5 and → 3. The 5 th dimension rolls into a circle of radius r 5

31. with (a miracle!) 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.