= P(a)*[P(o)]\\0) (2.48) that is, $[P(cr)]|0) is an eigenstate of the total momentum operator P(*. (2.60) The prime denotes the fact that the adjoint on string #3 is modified by a sign, so that, for example, if |0^ 3 )^ is some state built on |0;a 3) so that | A ^ ) = A^\\(f)^) where is some one-string operator (i.e. from Hi \u2014\u00bb Tii), then = (0 ( 3) |(-yl( 3 ) t) |2)|l) (2.61) whereas this sign is absent for strings #1 and #2, as ct\\ and cx2 are taken positive. This ensures the positivity of the free string energy 3 (H2) = (3|#2|2).|1) = J 2 e M H 2 r ) > 0- ( 2 - 6 2 ) r = l Below we will construct the cubic Hamiltonian and supercharges as states in 7i3. The subtlety (2.61) will only arise when considering operators in string #3's Hilbert space TL^f1. The vertex (2.59) respects the super-locality symmetry, but there is one last symmetry which we have yet to enforce. That is the commutation relations between the Q ' s given on the second line of (2.40). We see from (2.40) that Q~ and Q~, like H, commute with P1 and Q+ to give kinematical generators. Therefore we can build Q3 and Q3 using the progenitor vertex |V) as well. Recall that we included a prefactor h3 in our definition (2.50), which we then set to 1 and forgot about. The idea is to now restore this prefactor (and similar q3 and q3 for the supercharges) via an operator acting on \\ V) \\H3) = h3\\V), \\Q-)=q-\\V), \\Q-) = q-\\V) (2.63) and then to determine the specific form of the prefactors by ensuring the closure of the supersymmetry algebra (2.40). This process is simplified in the (2.38) basis for the fermions. There, as shown in appendix A , linear combinations of Q3 and Q3 may be taken so that {Q^,Q0lP2} = -2eaiPle^2H (2.64) i.e. we can factor away the dependence on J l J and J l ' J ' in (2.40). At first order in K, we will have schematically {Q2,Q3} ~ H3. Written using the state language, as in (2.63), we have zZ^r)aia2\\Q30102) + E Q{r)^2\\Qs a, d2) = -^ai01\u20ac&2^\\H3) , (2.65) 3 3 E^WdiaalQaAft) + Yl^(r)pl(32\\Q3aia2) = -^&1p^a202\\HZ) , (2.66) r=l r = 1 3 3 X ^ M ^ I Q s \/ W +J2Q(r)$1p2\\Q3a1a2) = 0 (2.67) where Q^r)p1p2 a n d Q(r)Pip2 a r e the quadratic, free string supercharges Q2. Finding a solution for the prefactors which obeys these relations is a non-trivial (and non-unique) undertaking. Chapter 2. Light-cone string held theory on the plane-wave 56 We will not step the reader through this process, and instead refer to the literature [73], where the following result is obtained \\H3) = g2 f(pa3, (K^ - - (KvKr -C t 3 O W 3 L pK ~o7 pK - K^K^saia2(Y)sl1&2(Z) - K^K\u00ab2\u00ab2s*ai0l2{Y)s^2(Z)\\\\V) 2 ( W Z ) W ^ \\Q3 3102) = 92Vf(pa3, r~)-^i3 a3 4 a'K '7171 -rz S \/ 3 l 7 2 (y) t* 2 . 2 (z)^) | i \/ ) , 0:3 4 ojf '7171 (2.68) ;172(^)w(^)^272)iv). where K = a\\a2a3) K', i f 7 are expressions linear in bosonic oscillators and are defined in (2.82), while Y and Z are their fermionic counter-parts and are given in (2.83). Also note that K, the coupling constant, has been replaced by g2 (2.16). The string coupling must be this value according to the A d S \/ C F T correspondence; it cannot be fixed by first principles, hence it is a matter of choice to set K = g2. Further ^7171 = X'a^111 , Lp212 = K V a i n 2 1 2 j f 7 i 7 i = Ki-(j-i't\/1'yi ( ^ 7 2 7 2 = J^i'pi'l\u2122 (2.69) where the (\/-matrices are defined in appendix A . We also have l + ^ + z V ^ z 4 I ~ 2 yi'f = Si'f i ~~ 2 Y\u2122(1 + \u00b1-Z4) - Z2ij(l + \u00b1-Y4)} + \\[Y2Z2]ij i - \u2014 (y4 + z4) + \u2014 r 4 z 4 iov ; 2.44 12 y a ^ ' ( i _ J_z4) - z 2 i Y ( i - -y 4 ) l + I [y 2 z 2 r Y v 12 7 v 12 ; J 4 L J Here we defined y W \u2014 _ U v2\u00abi \/3 l Z\u2122 =al3;Z 2 v 2 \\ i i _ y2fe(\u00ab2'2i)fc (y2z2) (2.70) and analogously for the primed indices. We have also introduced the following quantities quadratic and cubic in Y and symmetric in spinor indices v 2 = y y Q 2 1 aiffi \u2014 1OL\\OL21 Bi \u00bb V 2 = V y Q i (2.71) y 3 _ -y2 yPi y 2 y a 2 (2.72) Chapter 2. Light-cone string field theory on the plane-wave 5 7 and quartic in Y and antisymmetric in spinor indices 1 1 where Y 4 SE y a 2 l f t y- 2 a i f t = -YUY2a2*2 \u2022 (2-74) The spinorial quantities s and t are defined as s{Y) = Y+'-Y3 ,'. t ( Y ) = e + zT 2 - \\YA . ( 2 . 7 5 ) o o Analogous definitions can be given for Z. The normalization of the dynamical generators is not fixed by the superalgebra at 0(g2) and can be an arbitrary (dimensionless) function filets , ^) of the light-cone momenta and fj, due to the fact that P+ is a central element of the algebra. ' The definitions of the quantities Y, Z, K, and K, along with the bosonic and fermionic Neumann matrices are most easily expressed in the so-called \" B M N basis\" for the oscillators V2ai = ai+ e \/ n , iV2a*_n = o\u00a3 - a\u00a3n , V2b\u00abia2 = ffi1*2 + P^na2 , i\\fW}n2 = Pn.a2 - P-T > iy\/2b*l&2 = -(3^2 + 0t\\?2 , V2btT = P^2 + PtT ( 2 . 7 6 ) for n > 0 , and a*=a j a0' = a 0 ' 6g\"\u00bb = P^\"2 V>\u2122 = ft6* ( 2 . 7 7 ) for n = 0 . The commutation relations for the oscillators are then K , <#] = SmnS* , {{Pm)aia2, = < W \u00ab . ( 2 . 7 8 ) In order to perform the string-field theory calculations we are interested in comparing to gauge theory, we require the large-p limit of all quantities. These were worked out in [ 6 9 ] and are given in appendix B. We find simpler expressions for them, which are summarized in the B M N basis as \u20223 \\V) = \\Ea)\\Ep)S(^ar) ( 2 . 7 9 ) r=l where \\Ea) and \\Ep) are exponentials of bosonic and fermionic oscillators respectively i ^ H e * p u E E ) i a > 1 2 3 ( 2 ' 8 \u00b0 ) \\ r,s=lm,n=-oo \/ and Chapter 2. Light-cone string held theory on the plane-wave 58 ( 3 oo \\ r,s\u2014l m , n = \u2014 o o \/ where |aj)i23 = |0; OJI) Cg> |0; a 2) \u00ae |0; 0:3). We further have that 3 3 . s = l n S Z s = l n e Z = \u00a3 \u00a3 C ? w w ^ p , Z ^ = \u00a3 ^ G W ( s ) \/?l jf 2 , (2.83) s = l n 6 Z s = l n g Z where the large-p; limits of these quantities are repackaged from the expressions found in appendix B and are expressed as 2 fi*r = s in (n7rr )v^ : (A+A+ + A ^ A - ) ~rs = Vm (A+A+ + KAP) ; ( 2 > G 4 ) n q 2ny\/u^Jq~(q - Prn) ' q p 4ny\/uJqTJp~ (Bsuq + Brup)' A 3 r _ ?sin(|n|7rr) (o;g + Brun) ~ s _ % (Bsq - Brp) 2-K^\/unu>q (q - Brn) ' q p An^\/uqujp [Bsuq + Bru>p) where Qsnrq = QsJq \u2014 Qrqsn, Br = \u2014ar\/a3 for r = 1, 2, and where we remind the reader that 0:3 < 0 while a-[,a2 > 0. Also r = Bx while B2 = 1 \u2014 r. The mode number n is associated with string 3, while p and q are used for either string 1 or 2. We also drop the string label on Uq, Kq, Gq etc. as it is obvious from the quantity given. For example uiq in N%q should (r) \/~c be understood as . Continuing, we also have M l - r) A\" - A + Kn = +a3sm(nnr)J { ] \" (2.86) V na' K , - - J ^ \u00a3 ^ L , (2.87) G . - - ^ = , Gn = - ^ t l (2.88) where A+ = y\/uq- Brjia3, Aq = e{q)y\/uq + pr(j,a3, (2.89) A+ = y\/un - p,a3, A~ = e{n)y\/un + u.a3. (2.90) 2These expressions (2.84-2.88) are also valid for q,p = 0, except in the case of NQ\u00a7 = \u2014 N\u2122\\q,p=o, and in the case of Qgg = -ir_1^rQ^p|n,P=o- , Chapter 2. Light-cone string held theory on the plane-wave 59 2.2.4 The contact interaction Consider the commutation relation (2.64) at the next order in the coupling constant - i.e. 0 ( g r , ) , we have { Q 2 a i d 2 > Q&fofc} + { $ 4 a i d 2 > Qifiifc} { ^ ? 3 a i d 2 i Qzfofc} = _ 2 e a l \/ 3 i e d 2 ^ 2 - ^ 4 \u2022 (2.91) Determining Q4 has been a long sought-after but yet to be realized undertaking since the early days of light-cone string field theory on flat space [41-44]. Since it remains to be determined, the solution in the plane-wave case has been to simply set it to zero. This is a self-consistent choice which gives rise to the so-called contact interaction (see appendix A) HA = -Q3ia2Q3aia2- (2.92) In flat space [41-44], the 2 \u2014> 2 string process requires a contribution from Q4 to close the algebra (2.91). Here, we will be concerned with the plane-wave 1 \u2014> 1 string process; specifically the one-loop mass shift depicted in figure 2.7. Here two H3 vertices alternately split and then rejoin the strings at separated light-cone times, while the contact interaction coalesces the splitting and joining to a single event (the moment of contact). It was argued Figure 2.7: The one-loop process contributing to the shift of the energy or mass of a string state. Two H3 vertices may be combined to form a standard one-loop diagram. The contact interaction H4 (shown on the right) also contributes to this process. Unlike the first diagram, H4 acts at a single time, while each H3 acts at a different time; hence the name contact. in [70] that Q 4 cannot contribute to the 1 \u2014> 1 string process on account of it being quartic in string fields at tree level. Later, in analyzing the 1 \u2014* 1 string process on the plane-wave [75] argued less restrictively that although setting Q4 = 0 in this setting still allows the algebra to close, this is only a necessary but not sufficient condition. It is the opinion of the author that this issue has not been fully resolved; however the work in this thesis follows the fashion of setting this quartic supercharge to zero only because of the lack of another option. Determining the full expression for Q4 in the flat space or in the plane-wave background remains a potentially crucial element in the development of the light-cone string field theory. Chapter 2. Light-cone string field theory on the plane-wave 60 2.2.5 One-loop mass shift: impurity conserving channel We are now in a position to attempt the string theory calculation of the gauge theory result (2.15). The gauge theory result is valid for a general two-$ l-impurity operator (see (2.9), (2.12)), independent of the 50(4) x 50(4) representation (i.e. the spacetime index structure of the impurities). This allows for a choice of string state to consider for the calculation. Ideally, we would like to choose an 50(4) x 50(4) representation which can only be constructed out of bosonic oscillators. In this way, one circumvents having to worry about mixing between different string states of the same uncorrected energy. For example consider the representation which is scalar in both 5 0 (4)'s i[,i]> = \\: i.e. it can either be constructed out of two fermionic, or two bosonic oscillators (or to mirror the gauge theory discussion \"impurities\"). These two states have the same energy at g2 = 0, but when interactions are turned on, generically they will mix. To avoid this unpleasantness [75] used the following state3 | [9 ,\" l ]> ( y ) = ^= + c t fa l^ - ^ a J V - i ) |a> (2.94) whose 50(4) x 50(4) representation is unique. The one-loop mass shift proceeds using standard quantum mechanical perturbation theory 6E& = (4>n\\H3 ' { 0 ) V H3\\ n) (2.95) where |0 n ) represents the state whose shift we are calculating (i.e. (2.94)), which we take to be string #3 with uncorrected energy En\u00b0\\ and where V is a projection operator on the space of two-string states. Finally i \/ 2 n t is the free Hamiltonian (2.37), restricted to the internal strings 1 and 2. In practice it is not feasible to consider the full range of intermediate two-string states; instead, a cue is taken from the gauge theory computation where the total number of impurities contained in intermediate states is equal to that of the external state4. In the string theory computation, this is the so-called impurity-conserving channel, which is realized as follows (l + S\u00ab)5EnV = W ) ( [ 9 , l ] | g 3 JB^\u2022^KO.lD^ + ^ W ^ ^ l U g t i F Q a l l g . l ] ) ^ (2.96) En \u2014 r \/ 2 where5 v 3 The normalization of this state is 14-\\51^. One could have equally chosen |[1,9]); the string field theory would not produce a different result for the one-loop mass shift. 4 There is no reason for this logic to be extended to the string theory picture. Indeed, the results of the next section show that it is an unjustified truncation. 5Oscillators act only on the vacuum closest to them. Chapter 2. Light-cone string Geld theory on the plane-wave 61 p (2.97) 7 dr ( ls= E JQ 2 r ( l - r) (E0?*0^ ^ l 0 ^ ^ a V : + a J K a J L |a 2 )(a 2 | ao (ai| + alK fit* E 2 |a 2 )(a 2 | P* ^ \u2022 where r = \u2014ai\/a 3 so that 1 \u2014 r = \u2014a 2 \/ a 3 where we remind the reader that a 3 < 0, while a i | 2 > 0. The indices S i and S 2 are shorthand for indicating a sum over both dotted and un-dotted fermions. These projectors obey the condition 1% F = 1B,F, where we note further that the vacuua are normalized by . (ai|(a 2 |a 2)|ai) = r ( l - r), (a 3 |a 3) = 1. (2.98) Note that strictly we should have added a two fermion state to For the ([9,1]) state however, this contributes nothing as it requires a trace of the i,j indices. In order to calculate 5En2^, we will require the following matrix elements [75] W><[9, l]| (a 2 | a 0 (a'i| a0fc \\H3) = -2r(l - r) + ^ N*i N\u2122 A\u00ab\u00ab \\ a 3 \/ W)([9,1]| WMa^ct \\H3) = -2r(l - > ) ( ^ - ^f) A y w (2.99) a 3 a 3 r where we have used (C.10) and ( C . l l ) , and W)([9,1]| (a 2 | (Po)&l&2 (ai| a* |Q 3 \/ 3 l \/j 2> = -2i C G 0 ( 2 ) (ff\u201e (3) + K - n ( 3 ) ) < 0 A* V ) \u00a3 $ ^([9,l]|(a 2|(ai |(\/?_ pr^aJ|Q 3 \/ 3 l4 2> = -2i C (?,p|(1) ( ^ ( 3 ) i V i ; + \/v_ n ( 3 ) A?_ p ) A\u00ab V ) * <5g (2.100) where A\u00ab'H = {^*

(z) \u2014 G(z) exp(z5(z)); the leading function G(z) is of order 1, while the phase S(z), being proportional to the potential is of order J, Note also that in the plane-wave limit J ~ i? 2 . Keeping the leading terms only, the result of plugging the W K B form into (2.168) is z 2 f - R4z2u2 + J2 = 0, - \u2022 ^ = J&u*-\u00a3 (2.169) \\dz J dz V zA and therefore S(z) is real only if z2 > J2\/(u2R4); the boundary at z = 0 is obviously excluded. However, the field is free to tunnel to the boundary, i.e. we may let S(z) become imaginary. The tunnelling trajectory can then be found by noting

|2^ (2.170) and therefore associating the momentum along z to this via dz zz dr zl dr where r is the affine parameter along the trajectory. This gives (for the tunnelling solution) which integrates to z= m 3 , (2-173) Rzu) cosh r reproducing the catenary shown in figure 2.8. Replacing r \u2014> i r in (2.173) reproduces the Lorentzian geodesic (2.164) or equivalently corresponds to the propagating solution for (f>(z). However, note that since J ~ dip\/dr, where ip is an angle in S5, and ui ~ dt\/dr, where i is the time coordinate of the boundary C F T , such a rotation would need to be accompanied by a double Wick rotation of the AdS5 x S5 metric in which both t and ip become imaginary. This means that the tunnelling picture may be derived from the standard Lorentzian AdS$ x S5 through double Wick rotation of ip and t. ' In the< work [49], Dobashi and Yoneya continued this picture of holography to a con-struction of the light-cone string field theory vertices for the plane-wave background. They calculated the effective action for a massive scalar field along the aforementioned tunnelling trajectory. In their analysis, the first 50(4) excitations come directly from harmonic oscil-lator ground states where the frequency of the harmonic oscillator is given by the mass of the supergravity state (i.e. ra2 = A ( A \u2014 4)). The other 50(4), associated with D j Z in-sertions in the B M N operators, stem from excited states of these harmonic oscillators. The result is that the cubic coupling of the excited states is dictated completely by the cubic Chapter 2. Light-cone string Geld theory on the plane-wave 80 coupling of the ground states. This makes a definite prediction for the zero-mode sector of the string field theory; the cubic Hamiltonian H3 ought to only count excitations of the first 50(4). Of course this explicitly breaks the Z 2 symmetry of the plane-wave background. The perspective is that this symmetry is \"accidental\" from the point of view of holography, and indications that it is not manifest at the level of C F T three-point functions were discovered already in [79]. We will endeavour to give a concise summary of the construction of the D Y vertex. The effective action for computing the three-point functions of S U G R A scalars was worked-out in [32]. It is given by 4N2 r 0?Xy (2.174) and leads to agreement (via the G K P - W relation) with three-point functions of the dual C F T operators O&^x), where m2 = Aj(A, - 4), C 1 2 3 (2.175) \\x\\ - x2\\2a3\\x2 - x 3 | 2 a i | x 3 - Xi\\2a2 where cti \u2014 ( A 2 + A 3 \u2014 A i ) \/ 2 , and similarly for a2 and a 3 . The coefficient O i 2 3 is given by a Aj-dependent constant multiplied by the cubic coupling G i 2 3 C 1 2 3 = A ^ ( A 1 , A 2 , A 3 ) G 1 2 3 . (2.176) The strategy of Dobashi and Yoneya is to expand (2.174) about the tunnelling trajectory, quantize the free part of the action using creation\/annihilation operators, and then to cal-culate the matrix elements of the cubic Hamiltonian stemming from (2.1,74). Let r be the affine parameter along the tunnelling trajectory, while y = (x, z) are the fluctuations in the given coordinates (see (2.167)). The effective metric is then [49] ds2 = (l + f)df2 + df, while the free part of the effective action becomes ' - y2 u ~ T = T + \u2014 tanh r (2.177) \u2022 4 iV 2 (2vr)E dfd y 1 - \\V2) d&dfQi + dyQidyQi +(l + \\y2 ) A ; ( A ; - 4 ) $ ; ^ (2.178) where A = J + ki is the dimension of the B M N operator with k insertions of the JV = 4 S Y M scalar $V=4- Rewriting the fields as * 4 = e-Jft{QJ)m(T), $t = eJf 4>{QJ\\y)i>{r) where (j)0J\\y) is the ground state wave function of the operator \u2014 d2 + J2^ (2.179) (2.180) allows the y-directions to be integrated-out, leaving the following free action for the 4>{T) fields Chapter 2. Light-cone string held theory on the plane-wave 81 drZ~2 [^i^i ~ drAA + kitpiipi] (2.181) J ' i . \u2022 \u2022 and the following form for the interaction \\ f dl~ E ^ i . * a . < 3 ( ^ ^ 2 ^ 3 +h.c.) (2.182) where Xilti2,i3 = A4(Ai)Gili2i3, where A4(Aj) is a constant dependent on the A j , and we take J1 + J2 = J3 to conserve angular momentum. By comparing A123 with C123, a direct map may be made between the AdS5 x 5 5 couplings C123 and the (what ought to be) plane-wave cubic Hamiltonian coefficient A123. The result is that this coupling is proportional to k2 + k3 \u2014 k\\, i.e. the quadratic Hamiltonian counting the excitation energies of the B M N states A 1 2 3 oc k2 + k3 - ki. (2.183) The other SO(4) 's worth of excitations, corresponding to insertions of DiZ in the B M N operators, are conjectured to correspond here to excited states of 4>^ 4 ' \/ T \\ 1\/4 rf-m\/2 tt\\v) = II (- ^H^y\/Jyle-*'*, (2.184) where the excitation number n$ corresponds to the insertion of DiZ's in the B M N operator. The crucial element here is that the couplings for the excited states are related directly to those of the ground states via A^s 1 2 \" 3 = X\u00b0^J^j\\ J A ^ M t H M ? ^ ) - (2-185) The result is that the cubic Hamiltonian is still proportional only to the energies of the first 50(4) excitations k2 + k3 \u2014 k\\. At the end of the day, the vertex proposed by Dobashi and Yoneya must, at the level of supergravity states (i.e. string zero modes), count only the energies of the first 50(4). This is accomplished by taking an average of the Z2-even prefactor of D V P P R T and the Z 2 -odd prefactor of SVPS. In this way the second 50(4) zero modes cancel-out. The proposal is then l # 3 D Y > = 5 (|#3 D V P P R T> + | F 3 S V P S \u00bb \\ (2.186) IQ3DY) = \\ (IQ 3 D V P P R T ) + IQ3SVPS>.) Divergence cancellation The cancellation of divergences demonstrated in [77] for the SVPS vertex, was shown to extend to the D Y vertex in [78]. In fact, we now show that an arbitrary linear combination of the SVPS and D V P P R T vertices, Chapter 2. Light-cone string Geld theory on the plane-wave 82 H\u00bb = a tffVPS + p tf3DVPPRT (2.187) Qz = a Qly P S + \/? Q 3 D V P P R T (2.188) similarly yields a finite energy shift. We calculate the mass shift of the trace state as in section 2.3.1. The divergence stemming from the H3 term is simply a2 times the SVPS H3 divergence (2.113) plus (32 times the D V P P R T divergence (2.154). The reason is simple -the SVPS divergence stems from an entirely bosonic intermediate state, while (2.154) results from an entirely fermionic one. This precludes any divergences arising from cross-terms. We note that the SVPS divergence (2.116) is exactly equal to (2.154), therefore we have The pieces of the SVPS Q3 relevant to a two-impurity channel calculation are exactly Q D V P P R T W I T H K therefore, from (2.115) rj Ir (1 \u2014 r) a' - l 2 a 3 \/ 3 ) ^ 3 1 j _ ^ ( 3 ) ^ 3 ^ _ L \/ P ^ ( 3 ) i u 3 1 _ L ^ ( 3 ) ^ 3 1 ^l (\u201ek\\ 2Gg> ^ [a ( \/ v i 3 n i V ! n p . + tfM}) + P mNll + A ^ ) i V ^ p ) + 4 ( \/ ? ^ 1 ) + a ^ ) ) ^ 3 3 J a ^ ) ^ (2.190) The last term in (2.190) gives rise to a log-divergent sum, the large-p behaviour of which is 6E$ ~ +(a 2 + f) f dr &&Z*l (N^f \u00a3 i . (2.191) Thus, by the same arguments as section 2.3.1, the energy shift is finite for arbitrary a and (3. The D Y vertex uses a = 0 = 1\/2, and this combination exclusively gives rise to the agreement with gauge theory which will be presented in the next sub-section. Again, as for the D V P P R T vertex, the generalization of these arguments to the impurity non-conserving channels is a straightforward application of the treatment given in section 2.3.3. Impurity-conserving mass-shift In order to verify the validity of our results, we use two different methods for calculating the mass-shift. The first is straight-forward 5 E = (H^eHe^) + ^ Q r i e ) { e { Q r ) ( 2 1 9 2 ) where |e) is the |[9,1]) external state (2.94), and where the superscript \"int\" refers to internal states (i.e. strings number 1 and 2). For the second method, we recall.that Chapter 2. Light-cone string held theory on the plane-wave 83 J#3D Y> = \\ (0H2\\V) + \\H3)) |Q3DY> = \\(0Q2\\V) + | Q S \u00bb (2.193) where 9 = \u2014g2r(l \u2014 r)\/4> and \\H3) and |Q 3) are the SVPS vertices (2.68). Because of the simple form of the D V P P R T vertices, a perhaps simpler form for the D Y energy shift can be derived. We begin by considering some matrix elements = j r dx*pXN5 P(Xx)8f Q^apxR | ** ( t t ) ) ^ a \u201e\u00a3>\u201e + D\u201eDfj) - X-gp.v gpaDpa. (F.l) The action of DnDu on a scalar field **