Local SU(2) symmetry

13.1.1 The covariant derivative and interactions with matter In this section we shall introduce the main ideas of the non-Abelian SU(2) gauge theory which results from the demand of invariance, or covariance,

13.1. Local SU(2) symmetry 41

under transformations such as (13.2). We shall generally use the language of isospin when referring to the physical states and operators, bearing in mind that this will eventually mean weak isospin.

We shall mimic as literally as possible the discussion of electromagnetic gauge covariance in sections 2.4 and 2.5 of volume 1. As in that case, no free particle wave equation can be covariant under the transformation (13.2) (taking the isospinor example for definiteness), since the gradient terms in the equation will act on the phase factor α(x). However, wave equations with a suitably defined covariant derivative can be covariant under (13.2); physically this means that, just as for electromagnetism, covariance under local non-Abelian phase transformations requires the introduction of a definite force field.

In the electromagnetic case the covariant derivative is

D^μ = ∂^μ + iqA^μ(x). (13.4)

For convenience we recall here the crucial property of D^μ . Under a local U(1) phase transformation, a wavefunction transforms as (cf (13.3))

ψ(x)→ ψ^'(x) = exp(iqχ(x))ψ(x), (13.5) from which it easily follows that the derivative (gradient) of ψ transforms as

∂^μψ(x)→ ∂^μψ^'(x) = exp(iqχ(x))∂^μψ(x) + iq∂^μχ(x)exp(iqχ(x))ψ(x). (13.6) Comparing (13.6) with (13.5), we see that, in addition to the expected first term on the hand side of (13.6), which has the same form as the right-hand side of (13.5), there is anextra term in (13.6). By contrast, the covariant derivative of ψ transforms as (see section 2.4 of volume 1)

D^μψ(x)→ D^'μψ^'(x) = exp(iqχ(x))D^μψ(x) (13.7) exactly as in (13.5), with no additional term on the right-hand side. Note that D^μ has to carry a prime also, since it contains A^μ which transforms to A^'μ = A^μ −∂^μχ(x) whenψ transforms by (13.5). The property (13.7) ensured the gauge covariance of wave equations in the U(1) case; the similar property in the quantum field case meant that a globally U(1)-invariant Lagrangian could be converted immediately to a locally U(1)-invariant one by replacing

∂^μ by Dˆ^μ (section 7.4).

In appendix D of volume 1 we introduced the idea of ‘covariance’ in the context of coordinate transformations of 3- and 4-vectors. The essential notion was of something ‘maintaining the same form’, or ‘transforming the same way’. The transformations being considered here are gauge transformations rather than coordinate ones; nevertheless it is true that, under them, D^μψ transforms in the same way as ψ, while ∂^μψ does not. Thus the term covariant derivative seems appropriate. In fact, there is a much closer analogy between the ‘coordinate’ and the ‘gauge’ cases, which we did not present in volume 1, but give now in appendix N, for the interested reader.

13.1. Local SU(2)symmetry 43 where we have retained only the terms linear inEfrom an expansion of (13.8) withα→E. We have now dropped the x-dependence of theψ⁽¹²⁾’s, but kept that ofE(x), and we have used the simple ‘1’ for the unit matrix in the two-dimensional isospace. Equation (13.14) exhibits again an ‘extra piece’ on the right-hand side, as compared to (13.13). On the other hand, inserting (13.10) and (13.13) into our covariant derivative requirement (13.9) yields, for the left-hand side in the infinitesimal case,

D^'μψ^(12)'= (∂^μ+ igτ·W^'μ/2)[1 + igτ·E(x)/2]ψ⁽¹²⁾ (13.15) while the right-hand side is

[1 + igτ·E(x)/2](∂^μ+ igτ·W^μ/2)ψ⁽¹²⁾. (13.16) In order to verify that these are the same, however, we would need to know W^'μ – that is, the transformation law for the three W^μ fields. Instead, we shall proceed ‘in reverse’, and use the imposed equality between (13.15) and (13.16) to determine the transformation law of W^μ.

Suppose that, under this infinitesimal transformation,

W^μ→W^'μ=W^μ+δW^μ. (13.17) Then the condition of equality is

[∂^μ+ igτ/2·(W^μ+δW^μ)][1 + igτ·E(x)/2]ψ⁽¹²⁾

= [1 + igτ·E(x)/2](∂^μ+ igτ·W^μ/2)ψ⁽¹²⁾. (13.18) Multiplying out the terms, neglecting the term of second order involving the product ofδW^μ andE and noting that

∂^μ(Eψ) = (∂^μE)ψ+E(∂^μψ) (13.19) we see that many terms cancel and we are left with

igτ·δW^μ Using the identity for Pauli matrices (see problem 3.4(b))

σ·aσ·b=a·b+ iσ·a×b (13.21) this yields

τ·δW^μ =−τ·∂^μE(x)−gτ·(E(x)×W^μ). (13.22)

42 13. Local Non-Abelian (Gauge) Symmetries

We need the local SU(2) generalization of (13.4), appropriate to the local SU(2) transformation (13.2). Just as in the U(1) case (13.6), the ordinary

and differentiating term by term. By analogy with (13.7), the key property

1 us ‘decode’ the desired property (13.9), for the algebraically simpler case of an infinitesimal local SU(2) transformation with parameters E(x), which are

42 13. Local Non-Abelian (Gauge) Symmetries We need the local SU(2) generalization of (13.4), appropriate to the local SU(2) transformation (13.2). Just as in the U(1) case (13.6), the ordinary gradient acting on ψ⁽¹²⁾(x) does not transform in the same way as ψ⁽¹²⁾(x):

taking∂^μ of (13.2) leads to

∂^μψ⁽¹²⁾¹(x) = exp[igτ·α(x)/2]∂^μψ⁽¹²⁾(x)

+ igτ·∂^μα(x)/2 exp[igτ·α(x)/2]ψ⁽¹²⁾(x) (13.8) as can be checked by writing the matrix exponential exp[A] as the series

exp[A] = c∞ n=0

Aⁿ/n!

and differentiating term by term. By analogy with (13.7), the key property we demand for ourSU(2) covariant derivative D^μψ⁽¹²⁾ is that this quantity should transform likeψ⁽¹²⁾ – i.e. without the second term in (13.8). So we require

(D^1μψ⁽¹²⁾¹(x)) = exp[igτ·α(x)/2](D^μψ⁽¹²⁾(x)). (13.9) The definition ofD^μ which generalizes (13.4) so as to fulfil this requirement is

D^μ(acting on an isospinor) =∂^μ+ igτ·W^μ(x)/2. (13.10) The definition (13.10), as indicated on the left-hand side, is only appropri-ate for isospinorsψ⁽¹²⁾; it has to be suitably generalized for otherψ^(t)’s (see (13.44)).

We now discuss (13.9) and (13.10) in detail. The∂^μis multiplied implicitly by the unit 2 matrix, and the τ’s act on the two-component space ofψ⁽¹²⁾. TheW^μ(x) arethreeindependent gauge fields

W^μ= (W₁^μ, W₂^μ, W₃^μ), (13.11) generalizing the single electromagnetic gauge fieldA^μ. They are called SU(2) gauge fields, or more generallyYang-Mills fields. The term τ·W^μ is then the 2×2 matrix

τ·W^μ=

( W₃^μ W₁^μ−iW₂^μ W₁^μ+ iW₂^μ −W₃^μ

)

(13.12) using the τ’s of (12.25); the x-dependence of the W^μ’s is understood. Let us ‘decode’ the desired property (13.9), for the algebraically simpler case of an infinitesimal local SU(2) transformation with parametersE(x), which are of course functions ofx since the transformation is local. In this case,ψ⁽¹²⁾ transforms by

ψ⁽¹²⁾¹= (1 + igτ·E(x)/2)ψ⁽¹²⁾ (13.13) and the ‘uncovariant’ derivative∂^μψ⁽¹²⁾transforms by

∂^μψ⁽¹²⁾¹= (1 + igτ·E(x)/2)∂^μψ⁽¹²⁾+ igτ·∂^μE(x)/2ψ⁽¹²⁾, (13.14)

13.1. Local SU(2) symmetry 43

where we have retained only the terms linear in E from an expansion of (13.8)

1

with α → E. We have now dropped the x-dependence of the ψ^{( )}2 ’s, but kept that of E(x), and we have used the simple ‘1’ for the unit matrix in the two-dimensional isospace. Equation (13.14) exhibits again an ‘extra piece’ on the right-hand side, as compared to (13.13). On the other hand, inserting (13.10) and (13.13) into our covariant derivative requirement (13.9) yields, for the left-hand side in the infinitesimal case,

1 2

1

· W ^'μ/2)[1 + igτ 2

D^'μψ^{( )'} = (∂^μ + igτ · E(x)/2]ψ^{( )} (13.15) while the right-hand side is

1

· E(x)/2](∂^μ + igτ · W ^μ/2)ψ^{( )}2

[1 + igτ . (13.16)

In order to verify that these are the same, however, we would need to know W ^'μ – that is, the transformation law for the three W ^μ fields. Instead, we shall proceed ‘in reverse’, and use the imposed equality between (13.15) and

W ^μ (13.16) to determine the transformation law of .

Suppose that, under this infinitesimal transformation,

W ^μ → W ^'μ = W ^μ + δW ^μ . (13.17) Then the condition of equality is

1

[∂^μ + igτ /2· (W ^μ + δW ^μ)][1 + igτ · E(x)/2]ψ^{( )} 2 1

= [1 + igτ · E(x)/2](∂^μ + igτ · W ^μ/2)ψ^{( )}2 . (13.18) Multiplying out the terms, neglecting the term of second order involving the product of δW ^μ and E and noting that

∂^μ(Eψ) = (∂^μE)ψ + E(∂^μψ) (13.19) we see that many terms cancel and we are left with

τ · δW ^μ τ · ∂^μE(x)

ig = −ig

2 |(2 τ · E(x)) (τ · W ^μ ) (τ · W ^μ ) (τ · E(x))|

+ (ig)² − .

2 2 2 2

(13.20) Using the identity for Pauli matrices (see problem 3.4(b))

σ · aσ · b = a · b + iσ · a × b (13.21)

this yields

τ · δW ^μ = −τ · ∂^μE(x)− gτ · (E(x)× W ^μ). (13.22)

13.1. Local SU(2)symmetry 45 so thatψ⁽¹²⁾transforms by

ψ^(12)'=U(α(x))ψ⁽¹²⁾. (13.28) Then we require

D^'μψ^(12)'=U(α(x))D^μψ⁽¹²⁾. (13.29) The left-hand side is

(∂^μ+ igτ·W^'μ/2)U(α(x))ψ⁽¹²⁾

= (∂^μU)ψ⁽¹²⁾+U∂^μψ⁽¹²⁾+ igτ·W^'μ/2Uψ⁽¹²⁾, (13.30) while the right-hand side is

U(∂^μ+ igτ·W^μ/2)ψ⁽¹²⁾. (13.31) TheU∂^μψ⁽¹²⁾terms cancel leaving

(∂^μU)ψ⁽¹²⁾+ igτ·W^'μ/2Uψ⁽¹²⁾=Uigτ·W^μ/2ψ⁽¹²⁾. (13.32) Since this has to be true for all (two-component)ψ⁽¹²⁾’s, we can treat it as an operator equation acting in the space ofψ⁽¹²⁾’s to give

∂^μU+ igτ·W^'μ/2U=Uigτ·W^μ/2, (13.33) or equivalently

1

2τ·W^'μ= i

g(∂^μU)U⁻¹+U1

2τ·W^μU⁻¹, (13.34) which defines the (finite) transformation law for SU(2) gauge fields. Problem 13.1 verifies that (13.34) reduces to (13.26) in the infinitesimal case α(x)→ E(x).

Suppose now that we consider a Dirac equation forψ⁽¹²⁾:

(iγμ∂^μ−m)ψ⁽¹²⁾= 0 (13.35) where both the ‘isospinor’ components of ψ⁽¹²⁾ are four-component Dirac spinors. We assert that we can ensure local SU(2) gauge covariance by re-placing ∂^μ in this equation by the covariant derivative of(13.10). Indeed, we have

U(α(x))[iγμD^μ−m]ψ⁽¹²⁾ = iγμU(α(x))[D^μψ⁽¹²⁾)−mU(α(x)]ψ⁽¹²⁾

= iγμD^'μψ^(12)'−mψ^(12)' (13.36) using equations (13.9) and (13.28). Thus if

(iγμD^μ−m)ψ⁽¹²⁾= 0 (13.37)

)'

44 13. Local Non-Abelian (Gauge) Symmetries

Equating components of τ on both sides, we deduce

δW ^μ = −∂^μE(x)− g[E(x)× W ^μ]. (13.23)

The reader may note the close similarity between these manipulations and those encountered in section 12.1.3.

W ^μ Equation (13.23) defines the way in which the SU(2) gauge fields transform under an infinitesimal SU(2) gauge transformation. If it were not for the presence of the first term ∂^μE(x) on the right-hand side, (13.23) would be simply the (infinitesimal) transformation law for the T = 1 triplet repre-sentation of SU(2) – see (12.64) and (12.65) in section 12.1.3. As mentioned at the end of section 12.2, the T = 1 representation is the ‘adjoint’, or ‘regular’, representation of SU(2), and this is the one to which gauge fields belong, in general. But there is the extra term −∂^μE(x). Clearly this is directly analo-gous to the −∂^μχ(x) term in the transformation of the U(1) gauge field A^μ; here, an independent infinitesimal function Ei(x) is required for each compo-nent W _i μ (x). If the E’s were independent of x, then ∂^μE(x) would of course vanish and the transformation law (13.23) would indeed be just that of an SU(2) triplet. Thus we can say that under global SU(2) transformations, the W ^μ behave as a normal triplet. But under local SU(2) transformations they acquire the additional −∂^μE(x) piece, and thus no longer transform

‘prop-1

erly’, as an SU(2) triplet. In exactly the same way, ∂^μψ^{( )} 2 did not transform

‘properly’ as an SU(2) doublet, under a local SU(2) transformation, because of the second term in (13.14), which also involves ∂^μE(x). The remarkable

re-1

sult behind the fact that D^μψ^{( )} 2 does transform ‘properly’ under local SU(2) transformations, is that the extra term in (13.23) precisely cancels that in (13.14)!

To summarize progress so far: we have shown that, for infinitesimal trans-formations, the relation

1

2 1

(D^'μψ^{( )'}) = [1 + igτ · E(x)/2](D^μψ^{( )}2 ) (13.24) (where D^μ is given by (13.10)) holds true if in addition to the infinitesimal

1

local SU(2) phase transformation on ψ^{( )} 2 1

2 1

ψ⁽ = [1 + igτ · E(x)/2]ψ^{( )}2 (13.25) the gauge fields transform according to

W ^'μ = W ^μ − ∂^μE(x)− g[E(x)× W ^μ]. (13.26)

In obtaining these results, the form (13.10) for the covariant derivative has been assumed, and only the infinitesimal version of (13.2) has been treated explicitly. It turns out that (13.10) is still appropriate for the finite (non-infinitesimal) transformation (13.2), but the associated transformation law for the gauge fields is then slightly more complicated than (13.26). Let us write

U(α(x)) ≡ exp[igτ · α(x)/2] (13.27)

44 13. Local Non-Abelian (Gauge) Symmetries Equating components ofτ on both sides, we deduce

δW^μ=−∂^μE(x)−g[E(x)×W^μ]. (13.23) The reader may note the close similarity between these manipulations and those encountered in section 12.1.3.

Equation (13.23) defines the way in which the SU(2) gauge fields W^μ transform under an infinitesimal SU(2) gauge transformation. If it were not for the presence of the first term∂^μE(x) on the right-hand side, (13.23) would be simply the (infinitesimal) transformation law for theT = 1 triplet repre-sentation of SU(2) – see (12.64) and (12.65) in section 12.1.3. As mentioned at the end of section 12.2, theT = 1 representation is the ‘adjoint’, or ‘regular’, representation of SU(2), and this is the one to which gauge fields belong, in general. But there is the extra term−∂^μE(x). Clearly this is directly analo-gous to the−∂^μχ(x) term in the transformation of the U(1) gauge fieldA^μ; here, an independent infinitesimal functionEi(x) is required for each compo-nentW_i^μ(x). If the E’s were independent ofx, then ∂^μE(x) would of course vanish and the transformation law (13.23) would indeed be just that of an SU(2) triplet. Thus we can say that under global SU(2) transformations, the W^μ behave as a normal triplet. But underlocalSU(2) transformations they acquire the additional −∂^μE(x) piece, and thus no longer transform ‘prop-erly’, as an SU(2) triplet. In exactly the same way,∂^μψ⁽¹²⁾did not transform

‘properly’ as an SU(2) doublet, under a local SU(2) transformation, because of the second term in (13.14), which also involves∂^μE(x). The remarkable re-sult behind the fact thatD^μψ⁽¹²⁾ doestransform ‘properly’ under local SU(2) transformations, is that the extra term in (13.23) precisely cancels that in (13.14)!

To summarize progress so far: we have shown that, for infinitesimal trans-formations, the relation

(D^'μψ^(12)') = [1 + igτ·E(x)/2](D^μψ⁽¹²⁾) (13.24) (where D^μ is given by (13.10)) holds true if in addition to the infinitesimal local SU(2) phase transformation onψ⁽¹²⁾

ψ^(12)' = [1 + igτ·E(x)/2]ψ⁽¹²⁾ (13.25) the gauge fields transform according to

W^'μ=W^μ−∂^μE(x)−g[E(x)×W^μ]. (13.26) In obtaining these results, the form (13.10) for the covariant derivative has been assumed, and only the infinitesimal version of (13.2) has been treated explicitly. It turns out that (13.10) is still appropriate for the finite (non-infinitesimal) transformation (13.2), but the associated transformation law for the gauge fields is then slightly more complicated than (13.26). Let us write

which defines the (finite) transformation law for SU(2) gauge fields. Problem 13.1 verifies that (13.34) reduces to (13.26) in the infinitesimal case α(x)→

13.1. Local SU(2)symmetry 47 quantum number, so as to emphasize that it is not the hadronic isospin, for which we retain T; t will be the symbol used for the weak isospin to be introduced in chapter 20. The general local SU(2) transformation for a t-multiplet is then

ψ^(t)→ψ^(t)' = exp[igα(x)·T^(t)]ψ^(t) (13.42) where the (2t+ 1)×(2t+ 1) matricesT_i^(t) (i= 1,2,3) satisfy (cf (12.47))

[T_i^(t), T_j^(t)] = iEijkT_k^(t). (13.43) The appropriate covariant derivative is

D^μ=∂^μ+ igT^(t)·W^μ (13.44) which is a (2t+ 1)×(2t+ 1) matrix acting on the (2t+ 1) components of ψ^(t). The gauge fields interact with such ‘isomultiplets’ in auniversalway – only one g, the same for all the particles – which is prescribed by the local covariance requirement to be simply that interaction which is generated by the covariant derivatives. The fermion vertex corresponding to (13.44) is obtained by replacingτ/2 in (13.40) by T^(t).

We end this section with some comments:

(i) It is a remarkable fact that only one constantg is needed. This isnotthe same as in electromagnetism. There, each charged field interacts with the gauge fieldA^μvia a coupling whose strength is its charge (e,−e,2e,−5e . . .).

The crucial point is the appearance of the quadratic g² multiplying the commutatorof theτ’s, [τ·E,τ·W], in theW^μ transformation (equation (13.20)). In the electromagnetic case, there is no such commutator – the associated U(1) phase group is Abelian. As signalled by the presence of g², a commutator is a non-linear quantity, and the scale of quantities ap-pearing in such commutation relations is not arbitrary. It is an instructive exercise to check that, once δW^μ is given by equation (13.23) – in the SU(2) case – then theg’s appearing in ψ^(12)' (equation (13.13)) andψ^(t)' (via the infinitesimal version of equation (13.42)) must be thesameas the one appearing inδW^μ.

(ii) According to the foregoing argument, it is actually a mystery why electric charge should be quantized. Since it is the coupling constant of an Abelian group, each charged field could have an arbitrary charge from this point of view: there are no commutators to fix the scale. This is one of the motivations of attempts to ‘embed’ the electromagnetic gauge transfor-mations inside a larger non-Abelian group structure. Such is the case, for example, in ‘grand unified theories’ of strong, weak and electromagnetic interactions.

/

46 13. Local Non-Abelian (Gauge) Symmetries

FIGURE 13.1

Vertex for isospinor-W interaction.

then

1

(iγμD^'μ −m)ψ^{( )}2 ^' = 0, (13.38) proving the asserted covariance. In the same way, any free particle wave

1

equation satisfied by an ‘isospinor’ ψ^{( )} 2 – the relevant equation is determined by the Lorentz spin of the particles involved – can be made locally covariant by the use of the covariant derivative D^μ, just as in the U(1) case.

The essential point here, of course, is that the locally covariant form

in-1

2)’s and the gauge fields W ^μ, which are cludes interactions between the ψ⁽

determined by the local phase invariance requirement (the ‘gauge principle’).

Indeed, we can already begin to find some of the Feynman rules appropriate to tree graphs for SU(2) gauge theories. Consider again the case of an SU(2)

1

isospinor fermion, ψ^{( )}2 , obeying equation (13.38). This can be written as

1 2

1

∂ −m)ψ^{( )} = g(τ /2)· /W ψ^{( )}2 . (13.39) (i

In lowest-order perturbation theory the one-W emission/absorption process is given by the amplitude (cf (8.39)) for the electromagnetic case)

−ig { ψ¯⁽

f

1 2

1

) ( ) 2

·W ^μd⁴ x (13.40) (τ /2)γμψ_i

exactly as advertized (for the field-theoretic vertex) in (12.129). The ma-trix degree of freedom in the τ’s is sandwiched between the two-component

1

isospinors ψ^{( )}2 ; the γ matrix acts on the four-component (Dirac) parts of

1

2 The external W ^μ field is now specified by a spin-1 polarization vector ψ^{( )}.

E^μ, like a photon, and by an ‘SU(2) polarization vector’ a^r(r = 1, 2, 3) which tells us which of the three SU(2) W-states is participating. The Feynman rule for figure 13.1 is therefore

−ig(τ^r/2)γ_μ (13.41)

which is to be sandwiched between spinors/isospinors ui, u¯_f and dotted into E^μ and ar. (13.41) is a very economical generalization of rule (ii) in Comment (3) of section 8.3.1.

The foregoing is easily generalized to SU(2) multiplets other than doublets.

We shall change the notation slightly to use t instead of T for the ‘isospin’

46 13. Local Non-Abelian (Gauge) Symmetries

FIGURE 13.1

Vertex for isospinor-W interaction.

then

(iγμD^'μ−m)ψ^(12)'= 0, (13.38) proving the asserted covariance. In the same way, any free particle wave equation satisfied by an ‘isospinor’ψ⁽¹²⁾– the relevant equation is determined by the Lorentz spin of the particles involved – can be made locally covariant by the use of the covariant derivativeD^μ, just as in the U(1) case.

The essential point here, of course, is that the locally covariant form in-cludes interactions between the ψ⁽¹²⁾’s and the gauge fields W^μ, which are determined by the local phase invariance requirement (the ‘gauge principle’).

Indeed, we can already begin to find some of the Feynman rules appropriate to tree graphs for SU(2) gauge theories. Consider again the case of an SU(2) isospinor fermion,ψ⁽¹²⁾, obeying equation (13.38). This can be written as

(i/∂−m)ψ⁽¹²⁾=g(τ/2)· /Wψ⁽¹²⁾. (13.39) In lowest-order perturbation theory the one-W emission/absorption process is given by the amplitude (cf (8.39)) for the electromagnetic case)

−ig

{ ψ¯_f⁽¹²⁾(τ/2)γμψ_i⁽¹²⁾·W^μd⁴x (13.40)

exactly as advertized (for the field-theoretic vertex) in (12.129). The ma-trix degree of freedom in theτ’s is sandwiched between the two-component isospinors ψ⁽¹²⁾; the γ matrix acts on the four-component (Dirac) parts of ψ⁽¹²⁾. The externalW^μ field is now specified by a spin-1 polarization vector E^μ, like a photon, and by an ‘SU(2) polarization vector’a^r(r= 1,2,3) which tells us which of the three SU(2) W-states is participating. The Feynman rule for figure 13.1 is therefore

−ig(τ^r/2)γμ (13.41)

which is to be sandwiched between spinors/isospinorsui,u¯f and dotted into E^μanda^r. (13.41) is a very economical generalization of rule (ii) in Comment (3) of section 8.3.1.

The foregoing is easily generalized to SU(2) multiplets other than doublets.

We shall change the notation slightly to use t instead of T for the ‘isospin’

13.1. Local SU(2) symmetry 47

quantum number, so as to emphasize that it is not the hadronic isospin, for which we retain T; t will be the symbol used for the weak isospin to be introduced in chapter 20. The general local SU(2) transformation for a t-multiplet is then

ψ^(t) → ψ^(t)' = exp[igα(x)· T^(t)]ψ^(t) (13.42) where the (2t+ 1) × (2t+ 1) matrices T_i (t) (i= 1,2,3) satisfy (cf (12.47))

(t) (t) (t)

[T_i , T_j ] = iE_ijk T_k . (13.43) The appropriate covariant derivative is

D^μ = ∂^μ + igT^(t) · W ^μ (13.44) which is a (2t+ 1) × (2t+ 1) matrix acting on the (2t+ 1) components of ψ^(t). The gauge fields interact with such ‘isomultiplets’ in a universal way – only one g, the same for all the particles – which is prescribed by the local covariance requirement to be simply that interaction which is generated by the covariant derivatives. The fermion vertex corresponding to (13.44) is obtained by replacing τ /2 in (13.40) by T ^(t) .

We end this section with some comments:

(i) It is a remarkable fact that only one constant g is needed. This is not the same as in electromagnetism. There, each charged field interacts with the gauge field A^μ via a coupling whose strength is its charge (e,−e,2e,−5e . . .).

The crucial point is the appearance of the quadratic g2 multiplying the commutator of the τ ’s, [τ · E,τ · W ], in the W ^μ transformation (equation (13.20)). In the electromagnetic case, there is no such commutator – the associated U(1) phase group is Abelian. As signalled by the presence of g², a commutator is a non-linear quantity, and the scale of quantities ap-pearing in such commutation relations is not arbitrary. It is an instructive exercise to check that, once δW ^μ is given by equation (13.23) – in the SU(2) case – then the g’s appearing in ψ^{( 1 2 )'} (equation (13.13)) and ψ^(t)' (via the infinitesimal version of equation (13.42)) must be the same as the one appearing in δW ^μ .

(ii) According to the foregoing argument, it is actually a mystery why electric charge should be quantized. Since it is the coupling constant of an Abelian group, each charged field could have an arbitrary charge from this point of view: there are no commutators to fix the scale. This is one of the motivations of attempts to ‘embed’ the electromagnetic gauge transfor-mations inside a larger non-Abelian group structure. Such is the case, for example, in ‘grand unified theories’ of strong, weak and electromagnetic interactions.

Im Dokument Non-Abelian Gauge Theories (Seite 56-63)