8.324 Relativistic Quantum Field Theory II

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1 8.324 Reltivistic Quntum Field Theory II MIT OpenCourseWre Lecture Notes Hon Liu, Fll 200 Lecture 5.4: QUANTIZATION OF NON-ABELIAN GAUGE THEORIES.4.: Gue Symmetries Gue symmetry is not true symmetry, but reflection of the fct tht theory possesses redundnt derees of freedom. A ue symmetry implies the existence of different field confiurtions which re equivlent. For exmple, in the cse of U(), ψ e iα(x) ψ, A µ A µ + e µ α(x), the phse of ψ is not physicl deree of freedom. Similrly, nor is the lonitudinl prt of A µ. A mssless spin- representtion of the Lorentz roup hs only two polriztions. A µ hs four components. Thus to hve Lorentz covrint formultion, we require ue symmetries to et rid of the extr derees of freedom. physiclly inequivlent confiurtions ue orbits Fiure : Equivlent ue orbits in confiurtion spce. When quntizin the theory, we should seprte the redundnt nd physicl derees of freedom. We need to mke sure only physicl modes contribute to observbles. This leds to complictions in delin with ue theories. There re two enerl pproches:. Isolte the physicl derees of freedom: fix ue nd quntize the resultin constrined system. This method is used, for exmple, in xil ue quntiztion in quntum electrodynmics. 2. Retin the unphysicl modes, or even introduce dditionl modes, but mke sure tht they do not contribute to ny physicl observbles. This method is used, for exmple, in covrint pth interl quntiztion. For the first compliction in the pth interl quntiztion, consider, for exmple, the pth interl for sclr field Dϕ e dd x 2 ϕt Kϕ+V (ϕ) J T ϕ = e V ( δ δj ) e 2 d 4 xj T K J, () where K is the kinetic opertor ( 2 + m 2 ) nd K is the proptor for ϕ. For ue theories, the inverse of K is not defined. For exmple, in quntum electrodynmics, F µν F µν = ( µ A ν ν A µ )( µ A ν ν A µ ) = A µ K µν A ν + totl derivtives,

2 with K µν = 2 η µν µ ν. We see tht K µν ν α(x) = 0 for ny α(x), nd so the mtrix is sinulr. These zero eienmodes re the confiurtions which re ue-equivlent to 0. Non-Abelin ue theories hve the sme qudrtic kinetic terms s quntum electrodynmics. In order for K to hve n inverse, we need to seprte ue orbits with physiclly inequivlent confiurtions..4.2 Fdeev-Popov method: Exmple : A trivil exmple φ physiclly inequivlent confiurtions ue orbits Fiure 2: The rdil direction ives inequivlent confiurtions, nd the circles of fixed rdius re the ue orbits. Consider nd suppose f(x, y) only depends on r = x 2 + y 2. Then W = drdϕ re f(r) = 2π dr re f(r), W = dxdy e f (x,y) (2) where the 2π is the fctorized orbit volume. Equivlently, we cn insert delt function. More explicitly, we cn insert fctor of dϕ 0 δ(ϕ ϕ 0 ) =. Then W = dϕ 0 dxdye f(x,y) δ(ϕ ϕ 0 ). (3) The dϕ 0 intertes over the ue orbit, nd the other fctor intertes over section of non-ue equivlent confiurtions. Exmple 2: Gue theories We consider pure-ue theories only; ddin mtter fields is trivil. We define set of ue-fixin conditions: dim G Z = DA µ(x) e is[a µ]. (4) = f (A) = 0, =,..., dim G, (5) in order to select section of non-equivlent confiurtions: δf (A Λ (x)) = dλ (x)δ(f (A Λ )) det. (6) b (y) Here, the determinnt is the determinnt of both the color nd function spce. Insertin (6) into (4), usin n brided nottion, Z = dλ DA e is[a] δf(a Λ ) δ(f(a Λ )) det. (7) 2

3 Now we observe tht DA = DA Λ, s ue trnsformtions correspond to unitry trnsformtions plus shifts, nd tht S [A] = S [A Λ ], becuse of the definin ue symmetry. Hence, Z = dλ DA Λ e is[aλ] δf(a Λ ) δ(f(a Λ )) det = dλ DAe is[a] δf(a) δ(f(a)) det, s A Λ is dummy intertion vrible. So, in we fctor the prtition function into the ue volume dλ nd pth-interl over ue-inequivlent confiurtions which is independent of Λ. We redefine this ltter fctor to be the new prtition function; tht is, δf(a) Z DAe is[a] δ(f(a)) det. (8) Exmple 3: Axil ue From this, f (A) = A z = 0. f (A Λ ) = A z + ( z Λ + f bc A b zλ c ), nd hence, δf (A Λ (x)) b (x ) = z δ b δ(x x ). () We see tht the Jcobin is independent of A µ : its determinnt only ives n overll constnt in the prtition function. Hence, Z = DAe is[a] δ(a z ) (2) up to constnt. This is prticulrly simple form. However, the drwbck is tht Lorentz covrince hs been broken. In enerl covrint ue, both the determinnt nd delt-function fctors re more difficult to work with. We need to employ dditionl tricks. (i) Determinnt fctor (9) (0) Recll tht dψdψ e ψm bψ b = det M b, (3) where the ψ nd ψ b re independent Grssmn vribles. Hence, the Fdeev-Popov determinnt is iven by δf (A Λ (x)) det b (y) = DC (x)dc(x) e i d4 xd 4 δf(a y C(x) Λ(x)) C b (y), b (y) (4) where C (x) nd C (x), =,..., dim G, re rel fermionic fields with no spinor indices. These re the host fields. (ii) Delt-function fctor Ain, the method is to write this fctor in the form of n exponentil. Firstly, enerlize δ(f (A)) δ(f (A) B (x)) (5) where B (x) is n rbitrry function. This does not chne the Fdeev-Popov determinnt. Therefore, Z is independent of B (x), nd so we cn weiht the internd of Z with Gussin distribution of B (x). Tht is, Z = DB (x)e i d 4 x B 2 2ξ (x) δf(a) DAe is[a] δ(f(a)) det. (6) 3

4 Collectin (4) nd (6), we find for Z, Z = DADC D C e is ef f [A,C, C] (7) with the effective ction S eff iven by [ ] δf (A (x) Λ S eff [A] = S Y M [A] d x f (A) + d xd y C (x) C b (y). (8) 2ξ b (y) Exmple 4: Lorentz ue µ f(a µ ) = A µ. (9) We hve tht (A ) (x) = A cd c d µ Λ µ(x) + ( µ Λ (x) + f A µ Λ ), nd so the Jcobin is iven by ivin with δf (A Λ (x)) = µ δb µ + cb Aµ c δ (4) (x y), (20) b L eff = L [A] + L f + L h, (2) µ 4 L µ f = ( A µ ), L h = d x C (x) D µ C (x), (22) 2ξ where D µ C (x) µ C (x) + f bd A b µc d. From this, we cn derive the Feynmn rules for the theory. 4

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