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- 1. Gauge Field Theory August 20, 2013
- 2. Brief Outline 1. Gauge invariance in classical electrodynamics 2. Local gauge invariance in quantum mechanics 3. Yang-Mills theory
- 3. Gauge Invariance and Classical Electrodynamics • In classical electrodynamics, the electric and magnetic ﬁelds can be written in terms of the scalar and vector potentials B = × A E = − φ − ∂A ∂t • However, these potentials are not unique for a given physical ﬁeld. There is a certain freedom in choosing the potentials. • The potentials can be transformed as Aµ (x) → A µ (x) = Aµ (x) + ∂µ Λ(x) without affecting the physical electric and magnetic ﬁelds.
- 4. Local Gauge Invariance in Quantum Mechanics Charged Particle in the Electromagnetic Field • Hamiltonian of a charged particle moving in the presence of the electromagnetic ﬁeld is given by H = 1 2m (p − qA)2 + qφ • Quantum mechanically, the charged particle is described by the Schr¨odinger equation, − 1 2m − iqA 2 ψ(x, t) = i ∂ ∂t + iqφ ψ(x, t)
- 5. Local Gauge Invariance in Quantum Mechanics Gauge Invariance and Quantum Mechanics • Classically, the potentials φ and A are not unique for a given physical electromagnetic ﬁeld. • We can transform the potentials locally without affecting the physical ﬁelds (and hence the behaviour of the charged particle moving in the ﬁeld). • We want to investigate whether an analogous situation exists in quantum mechanics (i.e. whether quantum mechanics respects the gauge invariance property of electromagnetic ﬁelds)
- 6. Local Gauge Invariance in Quantum Mechanics • The gauge transformation of the potentials does not leave the Schr¨odinger equation invariant. • However, it is possible to restore the form invariance of the Schr¨odinger equation, provided the transformation of the potentials Aµ → A µ = Aµ + ∂µ Λ(x) is accompanied by a transformation of the wave function ψ → ψ = e−iqΛ(x) ψ • With these two transformations together, the form invariance of the Schr¨odinger equation is assured (i.e. A µ and ψ satisfy the same equation as Aµ and ψ.)
- 7. Local Gauge Invariance in Quantum Mechanics Summary Quantum mechanics respects the gauge invariance property of the electromagnetic ﬁeld. It gives the freedom to change the electromagnetic potentials but at the cost of a simultaneous change in the phase of the wave function.
- 8. Local Gauge Invariance in Quantum Mechanics Reversing the Argument (Demanding Local Gauge Invariance) • Instead of starting with the charged particle Schr¨odinger equation, we start with the free particle Schr¨odinger equation − 1 2m 2 ψ(x, t) = i ∂ ∂t ψ(x, t) • We demand that this equation remains invariant under the local phase transformation of the wave function ψ(x) → ψ (x) = e−iqΛ(x) ψ(x) • However, the new wave function ψ (x) does not satisfy the free particle Schr¨odinger equation.
- 9. Local Gauge Invariance in Quantum Mechanics • We conclude that the local gauge invariance is not possible with the free particle Schr¨odinger equation. • However, the demand of local gauge invariance can be satisﬁed by modifying the free particle Schr¨odinger equation. • It turns out that by modifying the derivative operators in the free particle Schr¨odinger equation as ∂µ → Dµ = ∂µ + iqAµ we can achieve the required goal, provided the vector ﬁeld Aµ also transforms under the phase transformation of the wave function ψ.
- 10. Local Gauge Invariance in Quantum Mechanics Summary • Local gauge freedom in the wave function in quantum mechanics is not possible with the free particle Schr¨odinger equation. • The insistence on the local gauge freedom forces us to introduce in the equation a new ﬁeld which interacts with the particle.
- 11. Yang-Mills Theory • We now turn to extend the concept of local gauge invariance to ﬁeld theories. • In ﬁeld theory, the quantity of fundamental interest is the Lagrangian density of the ﬁelds and accordingly, we demand the local gauge invariance of the Lagrangian density.
- 12. Yang-Mills Theory Lagrangian Density • We consider a Lagrangian density which depends upon the scalar ﬁeld φ and its ﬁrst derivative ∂µφ L ≡ L(φ(x), ∂µφ(x)) • We also assume that the Lagrangian density is constructed out of the inner products (φ, φ) and (∂µφ(x), ∂µφ(x)) (where bracket denotes the inner product in ﬁeld space), e.g. L = (∂µφ)† (∂µφ) − m2 φ† φ − λ(φ† φ)2 where the ﬁeld φ, in general, is a multi component ﬁeld.
- 13. Yang-Mills Theory (Inﬁnitesimal group theory) • We are mainly interested in the compact Lie groups such as SU(N) and SO(N). • One basic property of the compact groups is that their ﬁnite dimensional representations are equivalent to the unitary representation. • The advantage of unitary transformations is that they preserve the inner products φ† φ → (Uφ)† (Uφ) = φ† (U† U)φ = φ† φ
- 14. Yang-Mills Theory (Inﬁnitesimal group theory) • Associated with each Lie group is a Lie algebra. The elements ω of the group can be written as T(ω) = eiλaT(ta) where ta are the generators of the group and T is some representation. • One important representation is the adjoint representation for which the Lie algebra space coincides with the vector space on which the group elements act. The action is given by Ad(ω)A = ωAω−1 where, A is an element of the Lie algebra space.
- 15. Yang-Mills Theory (Global symmetry transformation) • We now assume that the Lagrangian density remains invariant under a global symmetry transformation φ(x) → φ (x) = T(ω)φ(x) where ω is an element of the symmetry group and T(ω) is some unitary representation under which the ﬁelds φ transform. • For example, the ﬁeld φ may be a two component object transforming under the fundamental representation of the SU(2) group, i.e. φ(x) ≡ φ1(x) φ2(x) → φ1(x) φ2(x) = eiΛaσa/2 φ1(x) φ2(x)
- 16. Yang-Mills Theory (Local symmetry transformation) • We now generalize the global transformation to a local transformation φ(x) → φ (x) = T(ω(x))φ(x) • Under a local transformation, the inner product φ†φ remains invariant. However, the inner product involving the derivative of the ﬁelds (∂µφ)†(∂µφ) does not remain invariant, since ∂µφ(x) → ∂µφ (x) = T(ω(x))∂µφ(x) + ∂µT(ω(x))φ(x) (The second term in the right hand side prevents the invariance of the inner product involving the derivatives)
- 17. Yang-Mills Theory (Introducing gauge ﬁelds) • To ensure the invariance of the Lagrangian density, the same procedure, as in the case of quantum mechanics, is followed. • We replace the ordinary derivative by a covariant derivative ∂µφ(x) → Dµφ(x) = (∂µ − igT(Aµ))φ(x) introducing a ﬁeld Aµ known as the gauge ﬁeld. • The ﬁeld Aµ is constructed in such a way that the covariant derivative transforms exactly as the ﬁeld φ, namely Dµφ(x) → (Dµφ(x)) = T(ω(x))Dµφ(x)
- 18. Yang-Mills Theory (Gauge ﬁelds belong to the lie algebra) • The last demand leads to the following transformation property for the gauge ﬁelds Aµ T(Aµ) = T(ωAµω−1 ) + i g T(ω∂µω−1 ) • Both the terms in the right hand side belong to the lie algebra of the corresponding symmetry group. • The ﬁrst term is a result of the action of adjoint representation. For the second term, we look at the group elements near identity ω(x) = 1 + iλa(x)ta + o(λ2 ) This gives ω∂µω−1 = −i(∂µλa)ta
- 19. Yang-Mills Theory (Gauge ﬁelds belong to the Lie algebra) • Since gauge ﬁelds Aµ belong to the Lie algebra space, it follows that we can write them as a linear combination of the generators ta Aµ = Aµ a ta • From this, it also follows that the number of independent gauge ﬁelds is equal to the number of generators of the group. Thus, e.g., if the symmetry group is SU(N), the number of gauge ﬁelds will be (N2 − 1). • Thus, the number of gauge ﬁelds depends only upon the underlying symmetry group and is independent of the number of matter ﬁelds present in the system (of course, the number of matter ﬁelds should match with the dimension of some representation of the symmetry group)
- 20. Yang-Mills Theory (Lagrangian density for the gauge ﬁelds) • Since we have introduced the gauge ﬁelds Aµ in our system, we need to have a term in the Lagrangian density which describes their dynamical behavior. • Moreover, this term should also be gauge invariant to preserve the gauge invariance of the Lagrangian density. • We recall that the electromagnetic Lagrangian density is given by L = − 1 4 FµνFµν where, Fµν = ∂µAν − ∂νAµ is the ﬁeld strength tensor.
- 21. Yang-Mills Theory (Lagrangian density for the gauge ﬁelds) • To construct the ﬁeld strength tensor for the gauge ﬁelds, we take guidance from the following theorem (Rubakov, chapter 3) “A Lie algebra is compact if and only if it has a (positive-deﬁnite) scalar product, which is invariant under the action of the adjoint representation of the group ” • Since our aim is also to have a gauge invariant term, we demand that the ﬁeld strength tensor for the gauge ﬁelds should also transform according to the adjoint representation, i.e. Fµν → Fµν = Ad(ω)Fµν = ωFµνω−1 and we construct the gauge invariant Lagrangian density using this ﬁeld tensor.
- 22. Yang-Mills Theory (Lagrangian density for the gauge ﬁelds) • This demand leads to the ﬁeld strength tensor Fµν = ∂µAν − ∂νAµ − ig[Aµ, Aν] • Since Fµν belongs to the Lie algebra, we can write it as a linear combination of the generators Fµν = Fµν a ta • In terms of the components Aµ a and Fµν a , we have Fµν a = ∂µ Aν a − ∂ν Aµ a + gfabcAµ b Aν c • This differs from the electromagnetic case by the presence of a non linear term.
- 23. Yang-Mills Theory (Lagrangian density for the gauge ﬁelds) • The Lagrangian density for the gauge ﬁelds is postulated to be the inner product Lgaugeﬁeld = − 1 2 Tr(Fµν Fµν) = − 1 4 Fµν a Faµν • Since Fµν transforms as the adjoint representation, this inner product is invariant (basically due to cyclic property of trace).
- 24. Yang-Mills Theory (Full Lagrangian density) • For the example given earlier, the complete Lagrangian density thus becomes L = (Dµφ)† (Dµ φ) − m2 (φ† φ) − λ(φ† φ)2 − 1 4 Fµν a Faµν where, Dµφ = (∂µ − igT(Aµ))φ
- 25. Yang-Mills Theory (Energy-momentum tensor) • The energy momentum tensor can be obtained using the deﬁnition δS = − 1 2 d4 x √ −g Tµν δgµν • This gives Tµν = 1 4 ηµν Fλρ a Faλρ − Fµλ a Fν a λ + 2(Dµ φ)† Dν φ − ηµν Lφ • Energy is given by integrating the (00)th component of this tensor over the spatial volume and is positive deﬁnite. E = d3 x (D0 φ)† D0 φ + (Di φ)† Di φ + m2 (φ† φ) + λ(φ† φ)2 + 1 2 F0i a F0i a + 1 4 Fij a Fij a
- 26. Yang-Mills Theory Summary The interaction between the scalar ﬁelds and the gauge ﬁelds can be obtained by invoking the local gauge invariance principle. This principle also dictates the kind of terms which can be present in the Lagrangian.
- 27. Why Yang-Mills Theory • Every physical phenomenon is believed to be governed by four interactions. Two of these, namely, Gravity and Electromagnetism are felt in day to day life. • Due to this, it is possible to formulate a classical version of these interactions. • The formulation of the universal Gravitational force law by Isaac Newton from the observation of the motion of an apple and the moon is an excellent example of this. • Similarly, the laws of Electrodynamics were discovered by observing the behavior of magnets, current carrying wires and so on. James Clark Maxwell gave the exact mathematical form of these laws using these observation (and his excellent insight).
- 28. Discovering Gravity Newton under the Apple tree
- 29. Discovering Electromagnetism
- 30. Why Yang-Mills Theory • The quantum version of the Electrodynamics (Quantum Electrodynamics) was constructed with the help of its known classical version. • However, there is no guidance in the form of classical laws for the strong and weak interactions. We have to directly deal with the quantum version of these interactions. • The Gauge invariance principle comes to rescue. The mathematical form of the strong and weak interactions has been constructed by using this principle.
- 31. References 1. Valery Rubakov, Classical Theory of Gauge Field, Princeton University Press, Princeton, New Jersey (2002) 2. Aitchison and Hey, Gauge Theories in Particle Physics: Volume 1, 3rd Ed., IOP (2004)
- 32. Thank You

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