aa r X i v : . [ h e p - t h ] J a n An Alternative to Collective Coordinates
Jarah Evslin and Hengyuan Guo
Institute of Modern Physics, NanChangLu 509, Lanzhou 730000, China andUniversity of the Chinese Academy of Sciences, YuQuanLu 19A, Beijing 100049, China
Collective coordinates provide a powerful tool for separating collective and elementary excitations,allowing both to be treated in the full quantum theory. The price is a canonical transformationwhich leads to a complicated starting point for subsequent calculations. Sometimes the collectivebehavior of a soliton is simple but nontrivial, and one is interested in the elementary excitations. Weshow that in this case an alternative prescription suffices, in which the canonical transformation isnot necessary. The use of a nonperturbative operator which creates a soliton state allows the theoryto be constructed perturbatively in terms of the soliton normal modes. We show how translationinvariance may be perturbatively imposed. We apply this to construct the two-loop ground state ofan arbitrary scalar kink.
When a theory is reformulated in terms of collec-tive coordinates, some phenomena involving largenumbers of elementary quanta, such as plasmawaves, can be treated in perturbation theory [1].Two groups applied collective coordinates to quan-tum solitons in the fateful Summer of 1975, allowinga treatment of the scattering of quantum solitons. In[2], following the spirit of [1], the collective coordi-nates are related to the elementary fields by a canon-ical transformation. This transformation allows astraightforward quantization of the system. How-ever it comes with a price, the theory becomes rathercomplicated and power counting renormalizability islost. Nevertheless, the authors are still able to studya soliton in motion and in Refs. [3, 4] the two-loopcorrection to a soliton energy is reproduced. A func-tional integral formulation is employed to avoid thecomplications of quantum states.In [5] collective coordinates are introduced with-out the canonical transformation. Following [1], theformulation was Hamiltonian. As no transformationwas used, the authors are forced to quantize the the-ory without collective coordinates to determine theoperator ordering in the theory with collective coor-dinates. The theory is still “considerably more com-plex than that usually encountered in quantum fieldtheory” but is now simple enough that the authorscan treat two soliton scattering. Due to the complex-ity of both approaches, quantum states were never considered beyond one loop, where the theories aresums of uncoupled quantum harmonic oscillators.Sometimes one is not interested in the collectiveexcitations. For example, one may be interested inthe quantum structure of a soliton in its rest frame.After all, intuition from large N [6] suggests thathadrons are quantum solitons and so their masses,form factors and general matrix elements may becalculated by solving for the corresponding quantumstate. In this case, we will propose a much simpleralternative to collective coordinates which allows oneto pass to higher numbers of loops using reasonablyelementary computations.For concreteness we will describe our formalism[7, 8] in the case of a real scalar field theory in 1+1dimensions, described by the Hamiltonian H = Z dx H ( x ) (1) H ( x ) = 12 : π ( x ) π ( x ) : a + 12 : ∂ x φ ( x ) ∂ x φ ( x ) : a + 1 g : V [ gφ ( x )] : a where :: a is the normal ordering defined below. Let φ ( x, t ) = f ( x ) (2)be a kink solution to the classical equations of mo-tion. We will always work in the Schrodinger pic-ture.We assume that V ′′ [ gf ( −∞ )] = V ′′ [ gf ( ∞ )] anddefine M / V with re-spect to it argument.As (2) is a solution of the classical equations ofmotion, we might be tempted to expand the quan-tum field as φ ( x ) = f ( x )+ η ( x ). Then φ → η = φ − f would be a passive transformation of the fields. Af-ter this transformation, the quadratic part of theHamiltonian H [ η ] would describe small perturba-tions about the classical kink, and one could proceedperturbatively.Instead of this passive transformation of the fields,following the standard approach [9, 10], we will con-sider an active transformation of the functionals act-ing on the fields. In particular, we transform theHamiltonian H [ φ, π ] → H ′ [ φ, π ] = H [ f + φ, π ] . (3)Below we will perform the same transformation onthe momentum operator P . The new observationthat lies behind our approach is that H ′ and H areunitarily equivalent, because H ′ = D † f H D f (4)where we have defined the translation operator D f = exp (cid:18) − i Z dxf ( x ) π ( x ) (cid:19) . (5)In general Eq. (4) will be applied to the regular-ized and renormalized H and will be our definitionof the regularized and renormalized H ′ . This elimi-nates the need to separately regularize H ′ and thento guess the correct regulator matching condition toapply when both regulators are taken to infinity. Ithas long been known [11] that the dependence onthe unknown matching condition leads to wrong an-swers in otherwise correct calculations. In (1) all UVdivergences are removed by the normal ordering, butthis choice was not necessary for our approach.The unitary equivalence (4) implies that H and H ′ have the same spectrum. Therefore the vacuumand the kink ground state are eigenstates of both Hamiltonians, with the same eigenvalues. We arethen free to use the vacuum Hamiltonian H to cal-culate the vacuum energy and the kink Hamiltonian H ′ to calculate the kink ground state energy. Wewill argue that this choice allows both calculationsto be performed in perturbation theory.This procedure will give us not only the energies ofthe kink states, but also the kink states themselves.Once an eigenstate of H ′ is found, one need onlyapply D f to arrive at the corresponding H eigen-state. For example, if | i is the eigenstate of H ′ corresponding to the kink ground state, then D f | i is the corresponding eigenstate | K i of H .This correspondence works already at tree level.Let | Ω i be a free vacuum of H that satisfies h Ω | φ ( x ) | Ω i = 0 . (6)This can be arranged by shifting φ by a constant.Then D † f | Ω i is the free vacuum as an eigenstate of H ′ .On the other hand, | Ω i is not eigenstate of H ′ ,or even of its free part. However it has a vanishingform factor (6) which one may expect for a tree-level vacuum. The corresponding state D f | Ω i in theeigenbasis of H is obtained via the unitary transfor-mation. As a result of (6) it has a form factor whichreproduces the classical kink profile h Ω |D † f φ ( x ) D f | Ω i = f ( x ) . (7)The state D f | Ω i is not the kink ground state | K i ,indeed it is not even an eigenstate of H just as | Ω i is not an eigenstate of H ′ . However it has the cor-rect form factor (7), leading one to suspect that thedifference between the two can be calculated in per-turbation theory as we now describe.We have argued that the eigenstate | i of H ′ cor-responding to the kink ground state is close to | Ω i .Our goal in this note will be to obtain a procedurewhich provides successively better approximationsto | i .The corresponding eigenstate of H will be | K i = D f | i . (8)The eigenvalue equation H ′ | i = Q | i (9)is easily solved at leading order as it reduces to afree theory and subleading orders can be solved bysimply fixing higher order coefficients, and so it isin principle possible to find an all-orders solution for | i . To obtain the correct eigenstate, we fix the lead-ing order energy to be minimal among eigenstates ofthe free part of H ′ . Had we not performed the uni-tary transformation, this program would have failedalready at the leading order, due to the inverse cou-pling appearing in the leading term in the solitonmass.To perform this perturbative calculation, we firstexpand H ′ in powers of the coupling H ′ = D † f H D f = Q + ∞ X n =2 H n (10) H = 12 Z dx h : π ( x ) : a + : ( ∂ x φ ( x )) : a + V ′′ [ gf ( x )] : φ ( x ) : a ] .Q is the classical kink mass and H n is order g n − .At one loop, only H is relevant. The constantfrequency ω solutions of its classical equations ofmotion are continuum normal modes g k ( x ) with ω k = √ M + k , discrete breathers and a Goldstonemode g B ( x ) = f ′ ( x ) / √ Q . Note that the definitionof ω k fixes the parametrization of k up to a sign.For brevity of notation, we will not distinguish be-tween continuum solutions and breathers, and so itwill be implicit that integrals over the continuousvariable k include a sum over the breathers, and 2 π times a Dirac delta function of continuum k shouldbe understood as a Kronecker delta of breathers.We choose the normalization conditions Z dxg k ( x ) g ∗ k ( x ) = 2 πδ ( k − k ) , Z dx | g B ( x ) | = 1(11)and conventions g k ( − x ) = g ∗ k ( x ) = g − k ( x ) , ˜ g ( p ) = Z dxg ( x ) e ipx (12) leading to the completeness relations g B ( x ) g B ( y ) + Z dk π g k ( x ) g ∗ k ( y ) = δ ( x − y ) . (13)As it is independent of time, the Schrodinger pic-ture field φ ( x ) may be expanded in any basis of func-tions even in the full, interacting theory. We willexpand it in terms of plane waves φ ( x ) = Z dp π (cid:18) A † p + A − p ω p (cid:19) e − ipx (14) π ( x ) = i Z dp π (cid:18) ω p A † p − A − p (cid:19) e − ipx and also normal modes [12] φ ( x ) = φ g B ( x ) + Z dk π (cid:18) B † k + B − k ω k (cid:19) g k ( x ) (15) π ( x ) = π g B ( x ) + i Z dk π (cid:18) ω k B † k − B − k (cid:19) g k ( x ) . Define the plane wave (normal mode) normal order-ing :: a (:: b ) by moving all A † (all φ and B † ) to theleft. The canonical algebra obeyed by φ ( x ) and π ( x )then implies[ A p , A † q ] = 2 πδ ( p − q ) (16)[ φ , π ] = i, [ B k , B † k ] = 2 πδ ( k − k ) . Decomposing fields in terms of the plane waveoperators, Bogoliubov transforming to the normalmode operators and then normal mode normal or-dering one finds that the one-loop Hamiltonian isa sum of quantum harmonic oscillators plus a freequantum mechanical particle for the center of mass H = Q + π Z dk π ω k B † k B k (17) Q = − Z dk π Z dp π ( ω p − ω k ) ω p ˜ g k ( p ) − Z dp π ω p ˜ g B ( p )˜ g B ( p )where Q is the one-loop kink mass. The one-loopkink ground state | i is therefore the solution of π | i = B k | i = 0 . (18)The whole spectrum may be obtained exactly at one-loop by creating normal modes with B † k and boostingwith e iφ k . The state | i is the first term in thesemiclassical expansion in powers of √ ~ | i = ∞ X i =0 | i i (19)where the n -loop ground state is the sum up to i =2 n − | K i at higher orders. As the one-loop spectrum is known exactly, the generalizationof what follows to other states is trivial. Recall that,using (8), it is sufficient to construct | i . | K i is an-nihilated by the momentum operator P = − Z dxπ ( x ) ∂ x φ ( x ) . (20)Therefore | i is annihilated by its unitary transform P ′ = D † f P D f = P − p Q π . (21)As g has dimensions of [action] − / , the quantity g ~ / is dimensionless. Setting ~ to unity, the semi-classical expansion in ~ is therefore equivalent to anexpansion in g . While P and π are independent of g , √ Q is proportional to g − and so ~ − . Thus theaction of P preserves the order in the semiclassicalexpansion while √ Q π reduces the order by one.Therefore (cid:16) P − p Q π (cid:17) | i = 0 (22)implies the recursion relation P | i i = p Q π | i i +1 . (23)Up to the kernel of π , this determines order i + 1states from order i states.We can now state the critical difference betweenour approach and the collective coordinate ap-proach. Whereas the collective coordinate approachimposes translation invariance exactly, we only solvethe recursion relation (23) up to the order at whichwe intend to find the state. As a result, no nonlinearcanonical transformation is required, only the linearBogoliubov transformation that relates the A p and B k . Thus we do not arrive at a complicated Hamilto-nian. On the contrary, perturbation theory is greatly simplified as we only need to solve for componentsin the kernel of π , the rest of the state is fixed bythe recursion relation. The momentum operator (20) is P = Z dk π ∆ kB (cid:20) iφ (cid:18) − ω k B † k + B − k (cid:19) (24)+ π (cid:18) B † k + B − k ω k (cid:19)(cid:21) + i Z d k (2 π ) ∆ k k (cid:16) − ω k B † k B † k + B − k B − k ω k − (cid:18) ω k ω k (cid:19) B † k B − k (cid:19) where we have defined the matrix∆ ij = Z dxg i ( x ) g ′ j ( x ) . (25)Integration by parts, using the fact that all g i ( x )vanish asymptotically, exchanges the indices and in-troduces a minus sign, so ∆ ij is antisymmetric. Wecan expand the i th order kink ground state as | i i = Q − i/ ∞ X m,n =0 Z d n k (2 π ) n γ mni ( k · · · k n ) × φ m B † k · · · B † k n | i . (26)Then the recursion relation becomes γ mni +1 ( k · · · k n ) = ∆ k n B (cid:16) γ m,n − i ( k · · · k n − )+ ω k n m γ m − ,n − i ( k · · · k n − ) (cid:17) +( n + 1) Z dk ′ π ∆ − k ′ B γ m,n +1 i ( k · · · k n , k ′ )2 ω k ′ − γ m − ,n +1 i ( k · · · k n , k ′ )2 m ! + ω k n − ∆ k n − k n m γ m − ,n − i ( k · · · k n − )+ n m Z dk ′ π ∆ k n , − k ′ (cid:18) ω k n ω k ′ (cid:19) γ m − ,ni ( k · · · k n − , k ′ ) − ( n + 2)( n + 1)2 m Z d k ′ (2 π ) ∆ − k ′ , − k ′ ω k ′ × γ m − ,n +2 i ( k · · · k n , k ′ , k ′ ) . (27)We have assumed here that γ i is symmetric under apermutation of the k j , but (27) yields a γ i +1 whichis not symmetric. Therefore, before each successiveapplication of the recursion relation, it is necessaryto symmetrize γ i +1 . The definition of the state (26)is invariant under this symmetrization.As is, the recursion relation applies to any kinkstate whose center of mass is at rest. To restrictto the ground state, we need only impose the initialcondition γ mn = δ m δ n γ . (28)One recursion yields γ ( k , k ) = ( ω k − ω k ) ∆ k k γ γ ( k ) = ω k ∆ k B γ . (29)Two yield the two-loop state up to the kernel of π , corresponding to γ n . These are reported inRef. [13].The terms γ n , which are in the kernel of π can befound using ordinary perturbation theory as follows.First define Γ to be any solution of i X j =0 (cid:16) H i +2 − j − Q i − j +1 (cid:17) | i j (30)= X mn Z d n k (2 π ) n Γ mni ( k · · · k n ) φ m B † k · · · B † k n | i where Γ i is of order O ( g i ). Recall that | i j is de-termined by γ j and so Γ is a function of γ . Thenobserve that the Schrodinger Equation( H − Q ) | i = 0 (31)is solved by any γ such thatΓ mni = 0 . (32)Recall that only the γ ni need be determined pertur-batively, as only they lie in the kernel of π . Theother components were already fixed by the recur-sion relation (23).To solve (30) we first note that H n = 1 n ! Z dxV ( n ) [ gf ( x )] : φ n ( x ) : a (33)where V ( n ) [ gf ( x )] is the n th derivative of g n − V [ gφ ( x )] evaluated at φ ( x ) = f ( x ). These are converted into normal mode normal orderedexpressions using the Wick’s theorem stated andproved in [14]. As normal mode normal orderedexpressions act simply on | i , one can easily useEq. (30) to write Γ in terms of γ . At each new order i , the γ i appear linearly and so the condition thatΓ i = 0 in (32) is uniquely solved for γ i .The usual IR problems associated to perturbationtheory in the presence of a continuous spectrum areresolved here by the momentum constraint (22), asthey are resolved in the case of the collective coor-dinate approach. As this perturbative calculation isstandard, it is reported in the companion paper [13].It yields a general formula valid for the energy of anyscalar kink at two loops Q = V II − Z dk ′ π | V I k ′ | ω k ′ − Z d k ′ (2 π ) (cid:12)(cid:12) V k ′ k ′ k ′ (cid:12)(cid:12) ω k ′ ω k ′ ω k ′ (cid:0) ω k ′ + ω k ′ + ω k ′ (cid:1) + 116 Q Z d k ′ (2 π ) (cid:12)(cid:12)(cid:0) ω k ′ − ω k ′ (cid:1) ∆ k ′ k ′ (cid:12)(cid:12) ω k ′ ω k ′ − Q Z dk ′ π | f ′′ ( x ) | where V I m ···I ,α ··· α n = Z dxV (2 m + n ) [ gf ( x )] I m ( x ) g α ( x ) · · · g α n ( x ) . (34)Here we have introduced the contraction factor I ( x )determined by [14] ∂ x I ( x ) = Z dk π ω k ∂ x | g k ( x ) | (35)and the condition that it vanish at infinity.The two-loop scalar kink mass was previously onlyknown in the Sine-Gordon case [3, 4]. There itwas derived from 13 UV divergent diagrams, whichcan be combined into five finite combinations. Ourterms are always each UV finite, as we have normal-ordered from the beginning. In the Sine-Gordon casethe terms in our energy formula are these five finitecombinations. Our formula on the other hand alsoapplies to kinks in many other models, such as φ n models.However, by finding the two-loop state, and notjust the mass, one can do much more. For example,it would be straightforward to calculate form factors[15] and matrix elements. This would allow, for thefirst time, a truly quantum approach to meson-kinkscattering [16–18], breather excitation, acceleration[19, 20] and more. ACKNOWLEDGEMENT
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