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Part III - Quantum Field Theory - Example Sheet 2 solutionsDated: 11/09DAMTP - University of CambridgebyRaquel H. RibeiroPlease send any corrections or comments to [email protected]1Quantisation of a set of HOsGiven the classical Hamiltonian for a string and the canonical commutation relations, one mayintroduce the creation and annihilation operators as followsˆan≡an=ωn2qn+i√2ωnpnandˆa†n≡a†n=ωn2qn-i√2ωnpn,(1)which in turn should verify the usual commutation relations. These should be consistent with thecanonical commutation relations. Indeed[an, am]=ωn2qn+i√2ωnpn,ωm2qm+i√2ωmpm==i2ωnωm[qn, pm] +i2ωmωn[pn, qm] = 0similarly=a†n, a†mandan, a†m=-i2ωnωm[qn, pm] +i2ωmωn[pn, qm] ==-i2ωnωm(iδnm) +i2ωmωn(-iδnm) =δnm,(2)as required. Inverting (1):qn=1√2ωan+a†nandpn=iωn2a†n-an.(3)Plugging into the expression for the Hamiltonian, we getH=+∞n=1ωn2ana†n+a†nanas required.We acknowledge the existence of a ground state|0in the theory, which is annihilated byanydestruction operator,an|0= 0,∀n.Using the non-vanishing commutation relations (2), werewrite the Hamiltonian asH=+∞n=1ωn22a†nan+δnn=+∞n=1ωna†nan++∞n=1ωn2δnn.(4)The last term is clearly divergent and it is due to the vacuum energy. Indeed, for the ground state0|H|0=+∞n=1ωn2δnn.1
Now, usually one is interested in differences of energy1; hence, one may define all energies withrespect to the vacuum energy.Applying the prescription known asnormal ordering, it thenfollows that:H:=+∞n=1ωna†nan.(5)To confirm the way the creation operator acts on states, we compute the following commutationrelation:H, a†n=+∞m=1ωma†mam, a†n=-+∞m=1ωma†m(-δnm)=ωna†n,where we have used the identity [A, BC] = [A, B]C+B[A, C]. This result is in agreement withthe fact that when the creation operator acts on a given state, it increases its energy by a factor ofω:H a†n|0=ωna†n|0.Also,1,2, . . . ,N|H|1,2, . . . ,N=0|(aN)N. . .(a1)1+∞m=1ωma†mama†11. . .a†NN|0==+∞m=1mωm,where we have used the fact thatamH an†p|0=aman†Han†p-1|0 +ωmaman†p|0= (p ωn)aman†p|0.2Canonical quantisation relationsFrom the Fourier decomposition of a real scalar field and its conjugate momentum in the Schr¨odingerpicture, one can get an expression for both the creation and the annihilation operators. Firstlyd3x e-iq·xφ(x)=d3p(2π)32Epd3xapei(p-q)·x+a†pe-i(p+q)·x==d3p2Epapδ(3)(p-q) +a†pδ(3)(p+q)==12Eqaq+a†-q,1except for situations which lie in a gavitational context where the energy-momentum tensor sources theEinstein equations.