Perturbative Approach to Time-Dependent Quantum Solitons

Recently we have introduced a lightweight, perturbative approach to quantum solitons. Thus far, our approach has been largely limited to configurations consisting of a single soliton plus a finite number of mesons, whose classical limit is an isolated stationary or rigidly moving soliton. In this paper, with an eye to soliton collisions and oscillons, we generalize this approach to quantum states whose classical limits are genuinely time-dependent. More precisely, we use a unitary operator, inspired by the coherent state approach to solitons, to factor out the nonperturbative part of the state, which includes the classical motion. The solution for the quantum state and its evolution is then reduced to an entirely perturbative problem.


Motivation
The present work is motivated by three questions.The first concerns oscillons.Oscillons are time-dependent classical solutions whose existence depends on the sign of a leading nonlinearity [1].Therefore, in a sense, they are present in half of all theories and these include many of phenomenological interest.Oscillons are generally [2,3,4] created by any violent event, such as a first order phase transition or soliton collision.They are unstable, yet in general they are the longest lived unstable excitations.As a result, at intermediate timescales after a violent event they dominate the dynamics.However, it is widely believed that the situation in the quantum theory is very different.Refs.[5,6] have argued that in the quantum theory oscillons experience a rapid decay into pairs of elementary quanta.Therefore, it is believed that once quantum corrections are considered, oscillons do not dominate the dynamics at any time.In Ref. [7], it was argued that one-loop corrections to the states might radically alter these conclusions.Yet such corrections have never been calculated.We aim to present a formalism which will allow the calculation of such corrections.
Second, it has long been known [8,9,10] that kink-antikink collisions lead to an interesting fractal pattern of escape windows.Needless to say, fractal patterns can be affected qualitatively by even arbitrarily small perturbations [11,12,13,14].Thus various authors have wondered throughout the ages just what effect the leading quantum corrections have on this resonance structure, although the first serious study of this question appeared only quite recently [15].These first two questions are rather straightforward applications.The third is more speculative.The Unruh effect in many ways resembles the two-particle decay claimed for oscillons in Refs.[5,6].It is therefore interesting, in our opinion to calculate the leading quantum corrections to Rindler space states and to see how they affect the radiation spectrum.Indeed, an intriguing paper [16] has recently suggested that the usual choice of states, on the future wedge, leads to a secular evolution, similarly to the choice of oscillon states in the above works, suggesting that such quantum corrections indeed arise automatically in the natural course of evolution.

Goals
To attack these problems, one first needs to choose a formalism for treating quantum states corresponding to classically nontrivial configurations.Several such formalisms are available.At the linear level there are powerful spectral methods [17] and also the classical-quantum correspondence of Refs.[18,19].However, we are interested in not only linear but also the higher order nonlinear effects.The usual tool of choice in this regime is the collective coordinate method of Ref. [20].However, after separating the collective coordinates, one needs a complicated canonical transformation to return to canonical coordinates, leading to an infinite number of terms in the Hamiltonian.An additional infinite number of terms is required [21] at the nonlinear level so that this transformation will leave the path integral measure invariant.In the end, all of these terms lead to a formalism which is powerful but also cumbersome.
We therefore plan to address all of these questions using the linearized soliton perturbation theory introduced in Refs.[22,23].This is a variant of the strategy in Ref. [24] which avoids a notorious problem [25] arising from separately regularizing the vacuum and nontrivial sectors by regularizing only once, and then using a unitary transformation that is equivalent to the coherent state construction of solitons [26], with the necessary [27] correc-tions, to relate the sectors.
So far this formalism has largely been applied to states that are obtained by quantizing time-independent solutions of the classical equations of motion.That is not to say that dynamics has not been studied, indeed kink-meson scattering amplitudes have been calculated exhaustively at the leading order [28,29], and also the elastic kink-meson scattering amplitude has recently been calculated [30].However, the soliton is heavier than the perturbative degrees of freedom by a factor of the inverse coupling 1/λ, and so the effect of these perturbative processes on the the soliton is suppressed by the coupling λ.As λℏ is dimensionless, the perturbative degrees of freedom vanish in the classical limit ℏ → 0. Thus classically there was no dynamics.
For the three motivating questions, we are interested in a different regime.We are interested in classical solutions that are themselves time-dependent, corresponding to a nonperturbative time dependence in the quantum field theory.
How can we treat a nonperturbative time-dependence using perturbation theory?We do it in the same way as we treat a nonperturbative soliton in perturbation theory: We use a nonperturbative unitary transformation to separate out the nonperturbative part.We refer to the transformed states and operators as the comoving frame.The introduction and study of this frame are the main results of this paper.

Summary
There is one case in which linearized soliton perturbation theory has already been applied to a classically time-dependent soliton.This is the case of the moving kink, treated in Ref. [31] and used to calculate form factors in Refs.[32,33].This was possible, without the introduction of the comoving frame, because it is related by a Lorentz transformation to a solution with no time-dependence.Therefore our strategy will be to use this example to learn how to construct the comoving frame in general, for solutions that may not be kinks at all.We begin in Sec. 2 with a quick review of the classical kink.We review the quantization of the stationary kink in Sec. 3. Then we quantize the moving kink in Sec. 4. We see the obstructions to a perturbative approach.Therefore, in Sec. 5 we introduce the comoving frame, in the case of the moving kink.Once we have understood the comoving frame in this example, the generalization to an arbitrary time-dependent solution of the classical equations of motion is clear, and so we present our general construction in Sec. 6.The reader uninterested in the intuitive derivation via the rigidly moving kink can skip straight to Sec. 6 to see our results.

Classical Kink
We believe that the results of this paper can be readily generalized to more phenomenologically interesting theories.However, we will work in a 1+1 dimensional theory of a scalar ϕ(x) and its conjugate π(x), described by the Hamiltonian Here λ is the coupling constant, V is a degenerate potential and we have included the usual normal ordering :: a which acts trivially in the classical theory, but eliminates all ultraviolet divergences in the quantum theory, which we turn to in Sec. 3. If one wishes to add more dimensions or fermions, then counterterms will be necessary to treat the corresponding divergences.
The classical equations of motion are where we have defined Putting these together yields the classical equation of motion

Stationary Classical Kink
Consider a static kink solution The corresponding classical energy is Let us multiply the rightmost expression in Eq. (2.5) by f ′ (x) and integrate, yielding where C is a constant of integration.Let V (±∞) = 0 (2.9) then (2.10) So we learn that the two terms in (2.6) are equal Consider a small perturbation ϕ(x, t) = f (x) + g(x)e ±iωt . (2.12) The linearized equation of motion is the Sturm-Liouville equation There are three kinds of solutions, classified by their frequencies ω.There is always a unique, real zero mode g B (x) with ω B = 0.For every real k there is a continuum normal mode g k (x) with ω k = √ m 2 + k 2 .Finally there may be discrete, real shape modes g S (x) with 0 < ω S < m.We will define the meson mass m momentarily.We will fix the normalizations of the normal modes using the completeness relation The sign of g B (x) is fixed using

Moving Classical Kink
The moving kink satisfies the equations of motion Its energy is 3 Quantum Kink at Rest The normal ordering in Eq. (2.1) is defined at the mass of the field ϕ(x) near one of the minima of the potential The values obtained at the minima on the two sides of the kink must be equal, or else the kink will begin to accelerate [34].

Ground State Kink
The defining frame is a choice of identification of the vectors in the Hilbert space, which we denote with kets, with the states in the quantum theory.In the defining frame, let |K⟩ be vector corresponding to the ground state kink at rest, in other words it is defined to be the Hamiltonian eigenstate with the lowest energy in the kink sector, which consists of states with a single kink and a finite number of mesons.
We may rewrite this state using the unitary displacement operator where |0⟩ is defined to be D † f |K⟩.In the defining frame, |0⟩ represents a state in the vacuum sector, which consists of states with finite numbers of mesons and no kinks.
In the defining frame, the energy of a state is the corresponding eigenvalue of the Hamiltonian H, while time evolution is generated by the action of H.The state |K⟩ is defined to be a Hamiltonian eigenstate We define the kink frame to be a different identification of the vectors in the Hilbert space with quantum states.In particular, a state called |Ψ⟩ in the defining frame is called D † f |Ψ⟩ in the kink frame.For example, the kink ground state is denoted by |K⟩ in the defining frame but in the kink frame we call it |0⟩.In summary, the vector |0⟩ represents a vacuum sector state in the defining frame but it represents a kink sector state in the kink frame.
The transition between the frames is a passive transformation, and so it transforms not only the coordinates on the Hilbert space, but also the operators acting on those coordinates.For example, it transforms the Hamiltonian into the kink Hamiltonian H ′ The ket |0⟩ is a kink Hamiltonian eigenstate Note that this follows from Eqs. (3.3) and (3.4), it is a purely algebraic identity independent of the frame which is used to identify |0⟩ with a physical state.
More generally, for any operator O acting in the defining frame, we may define an operator O ′ acting in the kink frame For example, the ground state kink is translation invariant, which in the defining frame means and, upon multiplying by D † f becomes the kink frame statement To evaluate this further, let us decompose the ground state kink rest mass and, following [24], decompose the Schrodinger picture fields in terms of the normal modes g i (x) that solve the Sturm-Liouville equation (2.13) with frequency where we adopt the conventions The symbol represents the sum of an integral over continuum modes k with a sum S over shape modes S.
Then, using Eq. ( 2.15), one may calculate where the first term translates the kink center of mass and the second translates the mesons.
There is a perturbative expansion in powers of such that Each successive term in |0⟩ and H ′ contains an additional power of √ λ while each term in Q contains a power of λ.In particular, Q 0 is of order O(1/λ) and so The left hand side is the momentum of the center of mass of the kink while the right hand side is minus the meson momentum.Of course, these two contributions to the momentum must cancel because we are working in the center of mass frame as P ′ |0⟩ = 0.

Excited Kink
A kink may be excited by adding a meson or shape mode, which we will refer to formally using the index k.In the kink frame this excited state corresponds to the ket vector |k⟩ and in the defining frame to the vector D f |k⟩.We demand that it is translation invariant It is also required to be a Hamiltonian eigenstate At leading order, in the kink frame, the corresponding ket vector is and so in the defining frame at leading order it is D f |k⟩ 0 .The translation invariance condition fixes the leading correction, up to a term in the kernel of π 0 , to be In other words, up to corrections of order O( √ λ) in the kernel of π 0 and arbitrary corrections of O(λ) In particular, this approximation to |k⟩ is sufficient to ensure translation invariance P ′ |k⟩ = 0 at leading order.
To evaluate the commutator term, let us expand In the case of continuum and shape modes Assembling the pieces As a result of the time independence of f (x), one has Therefore, time evolution of any kink frame vector |Ψ⟩ can be easily derived from the defining frame time evolution equation One finds So we learn that the kink Hamiltonian H ′ generates time evolution in the kink frame.
4 Moving Quantum Kink

Boost Operator
In the defining frame, we may boost any state using the boost operator In the kink frame this becomes where H ′ is the density of the kink Hamiltonian.

Moving Ground State Kink
Consider an eigenstate of the both the Hamiltonian H and also the momentum P .As they mix under boosts e −iαΛ He iαΛ = Hcoshα + P sinhα, e −iαΛ P e iαΛ = P coshα + Hsinhα ( a boost of this state will also be an eigenstate of both H and P , but the eigenvalues will change.
1 The Defining Frame Define the state to be the kink ground state boosted to a velocity v or equivalently a rapidity α =arctanh(v).Now the eigenvalues are As |K⟩ v is written in the defining frame, the time evolution is just 2 An Inertial Frame The inertial frame is defined by the passive transformation D f In the inertial frame, the moving kink state is represented by an eigenvector of the kink Hamiltonian and the kink momentum Note that |0⟩ v is not annihilated by P ′ , indeed it has an eigenvalue proportional to Q which is of order O(1/λ), potentially mixing various orders in perturbation theory when this condition is imposed.
To see that this is reasonable, let us try to understand the state We may expand Λ ′ in powers of the coupling Recall that the semiclassical limit requires α ≪ 1.Let us therefore take the limit α → 0 but we do not impose that α/ √ λ → 0. In this limit one finds The β 2 term leads to an x integral proportional to xg 2 B (x) which is odd, and so it vanishes.The linear term, when integrated over β, is proportional to α 2 , which we drop as we are working at linear order in α.In all, the Lorentz transformation is Our inertial frame ground state kink is then Now The Q 0 α term is of order O(1/λ) and it exactly reproduces the eigenvalue in Eq. (4.11).Therefore this nonvanishing eigenvalue is simply a result of the e iα √ Q 0 ϕ 0 coefficient in the inertial frame state.It can be factored out, and then the usual perturbation theory derivation of the states follows with P ′ annihilating the rest of the state.This is done automatically in the comoving frame.
5 The Comoving Frame

Definition
The comoving frame is defined using the passive transformation which shifts ϕ(x) and π(x) by the classical moving kink solution.It is sometimes convenient to regard t as a parameter, so that one comoving frame is introduced for each value of t.One can use the comoving frame in which t is equal to the time, but that choice is not necessary.
Consider a state that is represented by the ket |Ψ⟩ in the defining frame.In the comoving frame, this state is represented by the ket |Ψ⟩ (t) (5.2)

The Ground State Kink
In particular, in the comoving frame, we write the moving ground state kink as Using the Baker-Campbell-Hausdorff formula we may evaluate (5.4) Note that the last factor is an overall constant phase.This may be simplified using Thus the ϕ(x) term in (5.4) cancels the e iαΛ ′ 1 = e −iα √ Q 0 ϕ 0 term in e iαΛ ′ , reported in Eq. (4.17), up to corrections of order α 3 √ Q 0 , which we drop as higher powers of α arise at higher orders of perturbation theory.
Assembling these results, we find (5.6) and so in the comoving frame, the kink ground state corresponds to the ket (5.7)We see that all that remains at order O(1/λ) is a constant phase, which will be inconsequential and we will drop it from now on.

At order O(1/
√ λ) one finds a π(x) tadpole term that Lorentz contracts and moves the boosted kink.At time t = 0 it does not move the boosted kink.We are free to choose any time to be t = 0, as this choice simply corresponds to the choice of kink position in D f which does not affect the state.The contraction appears only at order O(v 2 / √ λ).

Evolution
Evolution in the defining frame is generated by H.We will now use this fact to learn how to evolve kets in the comoving frame.
Evolution in the comoving frame is described by The second term is easily evaluated, as it corresponds to evolution in the defining frame (5.9) To evaluate the first term on the right hand side of (5.8), use Combining these terms one finds where the comoving Hamiltonian is defined to be This comoving Hamiltonian evolves the state while changing the comoving frame parameter t so that it equals the time.
Again we decompose it in powers H of the coupling and evaluate it The first term is (5.16) Unlike Eq. (2.17), the result is not Q 0 γ, as a result of the f f − ḟ 2 term which negates the kinetic energy.While the commutator term in (5.8) makes H ′(t) 0 more complicated, it also exactly cancels the tadpole in (5.17) It does not contain terms at higher orders, so these are given by D f .For example, leaving the normal ordering implicit, (5.18)

Operators
Refs. [24,23] construct the spectrum of a stationary kink in the kink frame using kink frame operators.In this subsection, we will see how to map from these operators to the comoving frame of a moving kink, so that the old construction may be imported to the case of a moving kink.

Consider an operator
acting on a stationary kink state |Ψ⟩ in the kink frame.How does it act on the corresponding boosted kink in the comoving frame?
After the action of the operator, in the kink frame the state is represented by the vector which, in the defining frame, corresponds to Now we may boost it, yielding a new state which in the defining frame is In the comoving frame, before acting with the operator the boosted state was Acting with the operator it becomes the state (5.22) which in the comoving frame is represented by where f . (5.25) Using Eq. (5.6) this yields our master formula For any operator O ′ acting on a stationary kink in the kink frame, we have found the corresponding operator O ′v(t) that acts on the moving kink in the comoving frame.This formula can be used to migrate all results concerning a stationary kink to a moving kink.
For example, we know how to build the spectrum of a stationary kink using the operators B ‡ , B, ϕ 0 and π 0 in the kink frame, and so this map yields the corresponding construction for a moving kink in the comoving frame.
Recall that, choosing t = 0, the Lorentz-contracting π(x) terms are of order O(v 2 / √ λ) and so may be ignored at lower orders.Therefore, at these low orders, for any stationary operator O ′ in the kink frame of a stationary kink, the operator acts identically on the same kink state, but boosted and in the comoving frame.Conversely, for any operator O acting on a moving kink in the comoving frame, the operator acts identically on the kink in its rest Lorentz frame in the kink frame.We remind the reader that the two uses of the word "frame" are distinct.One refers to a Lorentz frame and the other to the identification between kets and states.

Fields
Let us apply the transformation (5.28) to the scalar field O = ϕ(x) acting in the comoving frame of a moving kink.The action of the field ϕ(x) in the comoving frame of the moving kink corresponds to that of in the kink frame of a stationary kink, where we evaluated the commutator using the explicit expression for Λ ′ 2 in Eq. (4.13).In other words, the operator φ(x) acting on a state of a stationary kink in the kink frame changes it to the same state, up to a boost, as one would get by acting on the same state but boosted with ϕ(x), using the comoving frame.
Similarly, acting with the operator π(x) in the comoving frame has the same effect on the state as acting on the stationary kink with in the kink frame.Again dropping terms of order O(v 2 / √ λ), the Lorentz contraction on the potential term can be dropped and at time t = 0 we find the Sturm-Liouville operator acting on the normal modes, and so These equations took a long time to derive, but in the Lagrangian formulation they resemble the usual Lorentz transformation where π(x, t) = φ(x, t) and the Heisenberg equations of motion are satisfied for the comoving kink Hamiltonian.
Recall that in the kink frame of the stationary kink, the operators B ‡ and B are creation and annihilation operators for mesons.The operators B ‡′ and B ′ that play the same role in the comoving frame of the moving kink1 are more complicated to write explicitly, as one needs to add a commutator with respect to Λ ′ 2 following the prescription in (5.27).Nonetheless, we may decompose our operators ϕ(x) and π(x) into B ‡ ′ and B ′ to learn how the fields are related to the operators that create and destroy mesons in the comoving frame.To do this, one recalls that ϕ(x) and π(x) in the comoving frame act identically to φ(x) and π(x) in the kink frame, which we may write in terms of ϕ(x) and π(x) using the results above, and then decompose into B ‡ and B as usual.Then, by definition, the action in the comoving frame is given by replacing these with B ‡ ′ and B ′ .
Let us run through that argument more slowly and concretely.Consider the operator ϕ(x).The operators themselves, and the relations between them, are independent of the frame.However it will be convenient to think of this operator acting on a moving kink state in the comoving frame.Now, setting O ′ = ϕ(x) in (5.27) we have defined another operator O ′v (t) .Let us give it another name This operator, acting on the moving kink in the comoving frame, changes the state identically to the operator ϕ(x) acting on the kink in its rest frame, using the kink frame.Similarly, we can also use (5.27) to construct π ′ (x) which acts in the comoving frame as π(x) acts in the kink frame.
We now have three operators: ϕ(x), ϕ ′ (x) and π ′ (x) which have simple interpretations in the comoving frame.How are they related?At leading order if α, (5.33) Running the same argument with π(x) instead of ϕ(x) one obtains (5.34) Our goal is not to write ϕ(x) and π(x) in terms of ϕ ′ (x) and π ′ (x), but rather to write them in terms of the operators B ′ and B ‡′ that destroy and create mesons and shape modes in the comoving frame.To do this, let us run the decomposition (3.10) through or master formula (5.27) to produce the comoving frame version Plugging this into Eqs.(5.33) and (5.34) we find What does this formula mean?The fields ϕ(x) and π(x) are always the same operators, independent of Lorentz frames and frames identifying states with kets.However, the operators that interpolate between eigenstates of the free Hamiltonian, creating and destroying mesons and translating the center of mass, do depend on the frame.We have now presented the decomposition of ϕ(x) and π(x) in terms of those operators in the comoving frame.It can be applied to perturbative calculations in the comoving frame, as the comoving Hamiltonian is a functional of ϕ(x) and π(x).
Notice that the expansion in the rapidity α is in fact an expansion in αxω k .Let ω k be of order O(m), corresponding to mesons that are relativistic but not ultrarelativistic.Physical effects such as loop corrections to the kink mass are indeed dominated by this range of k.Then this approximation is only valid at |x| ≪ 1/(αm).The width of the classical kink is 1/m, and so the approximation is valid over a range of 1/α classical kink widths.Recall that we are interested in nonrelativistic kinks, as perturbation theory is only expected to apply in this case, and so α → 0. Therefore this maximum distance can be larger than many other characteristic scales in the problem.

The Main Lesson from this Example
We are interested in generalizing these results to other time-dependent solutions.The key to this generalization will be the following observation.While our derivation was rather tortuous, the decomposition (5.36) can be simply derived as follows.One starts with the classical field theory solution for a small perturbation of a stationary kink in the inertial frame, which, as the kink is stationary, coincides with the kink frame ϕ(x, t) Here ϕ 0 , π 0 , B ‡ and B are c-number parameters defining the solution.In quantum field theory this could be interpreted as an interaction picture field.
Then Lorentz boost, yielding a new solution to the classical equations of motion (2.4), where we separate the perturbations from the boosted kink using the transformation ϕ(x, t) → The decompositions of the quantum fields ϕ(x) and π(x) follow from the rule applied to the classical solution (5.38).In particular, one finds Eq. (5.36) expanding to linear order in v.

The Ground State Revisited
In the kink frame of a ground state kink, the leading approximation |0⟩ 0 to the ground state |0⟩ is annihilated by π 0 and the operators B k .In the comoving frame of a moving kink, these operators correspond to and (5.42) One can check that π ′ 0 and B ′ −k indeed annihilate the leading order comoving frame ground state (1 + iαΛ ′ 2 ) |0⟩ 0 .

General States
There are two kinds of questions that one may ask.The first is the initial value problem, in which one starts with an initial state at time t = 0 and asks what state arises at time t.This can, in principle, be solved in the comoving frame as time evolution is generated by H ′(t) which has been constructed above.
One may also ask the spectrum.There are several distinct questions here.The eigenstates of H ′(t) in the comoving frame are time-independent in the comoving frame, in the sense that the ket that describes the state in the comoving frame is the same at all times.However, although the ket itself is time-independent, the state to which it corresponds depends on t.Correspondingly, the defining frame kets are time-dependent.The operator H ′(t) , except for scalar terms, begins at order O(λ 0 ) and so in principle this problem can be treated perturbatively.
On the other hand, one may ask about states that are truly time-independent, such as the breather Hamiltonian eigenstates in the quantum Sine-Gordon model.These are states that are eigenstates of the Hamiltonian H in the defining frame, and so by D In the case of the moving kink, the f term is of order O(v 2 / √ λ) and so, if v 2 ≪ O( √ λ), it can be treated perturbatively.
Dropping it, we find One may recognize this tadpole as the part of the momentum operator P ′ corresponding to the kink center of mass, which was mixed with the Hamiltonian by the boost.In particular, if the state was translation-invariant before the boost, so that it was annihilated by P ′ up to terms of order unity, then after the boost this term will be of order O(1) because √ Q 0 π 0 on such a state is P on the state, and P is of order O(1).Therefore, for such states, there are no terms in f of order the inverse coupling and we conclude that the spectrum of translation-invariant states can be found perturbatively.
Physically this is reasonable.Before the boost, translation-invariance means that one starts in a flat superposition of states with different kink positions x 0 .After the boost, evolving in time, the kink moves, and so x 0 shifts.However, if the initial wave function of x 0 was constant, so that all kink positions were weighted equally, then this shift leaves the state invariant, and so the boosted state remains time-independent.Of course, these are the kinds of states that we focused upon and often constructed in previous papers.
More generally, in the case of time-dependent solitons that are not simply moving but also have some internal dynamics, this tadpole term always appears, reflecting the fact that the quantum state changes as a result of the classical evolution.But also in this general case, we expect that this evolution is trivial if the quantum state is a flat superposition over the entire classical trajectory.This will be manifested by the fact that the tadpole term will annihilate the state, and so one need only solve the eigenvalue equation for the perturbative part of f .For example, in the case of breathers and oscillons, we expect to be able to obtain the spectra perturbatively if we restrict our attention to states that are flat quantum superpositions over the coherent states corresponding to each classical configuration that appears over a period of their evolution.

A General Time-Dependent Solution
Finally we are ready to take the lessons learned from the rigidly moving kink and apply them to a general, time-dependent solution of the classical equations of motion.

Perturbations
Consider a time-dependent classical solution ϕ(x, t) = f (x, t) of Eq. (2.4).Now let us consider small perturbations g(x, t) ϕ(x, t) = f (x, t) + g(x, t). (6.1) The classical equations of motion are 0 At linear order in g this becomes 0 We are not always interested in Lorentz-invariant theories.For example, kink-impurity scattering is phenomenologically rich and tractable with our methods, and yet the impurity necessarily breaks spatial translation symmetry and may even break time translation symmetry.
However, in the case of a Lorentz-invariant theory, two solutions g are always present.These are the spatial and temporal zero modes corresponding to a spatial translation of the solution or a displacement of the solution along its trajectory.Here c B and c T are normalization constants which may, at this point, be chosen freely.
The semiclassical treatment of solitons only is a reasonable approximation for heavy solitons, whose mass is much greater than any momentum scale.They are necessarily nonrelativistic, but in a Lorentz-invariant theory there will be solutions related by Galilean invariance.These correspond to the solutions Formally one may worry that the linear expansion may not be trusted at large |t|.However, for the purpose of quantizing a soliton, t can be chosen freely, as a shift in t corresponds to a shift in the choice of f (x, t) in the moduli space of solutions.
In addition to these three solutions, consider a set of solutions g k (x, t) where the index k will in general contain a continuum corresponding to real values of k and also discrete shape modes S.This set of solutions must be linearly independent, but also must span the possible initial perturbations.
In practice, one needs to simply search for solutions to (6.3) until these span the space of functions, throwing away solutions with linearly dependent initial data.

The Comoving Frame
The displacement operator is defined to be The state |Ψ⟩ (t) in the comoving frame is defined to be the same state as D (t) † f |Ψ⟩ in the defining frame.Now the arguments in Sec. 5 may be imported, as the derivations are generally identical.In particular, time evolution is generated by the comoving Hamiltonian while time-independent states are eigenstates of the inertial Hamiltonian D f .In particular Note that g B , g T and g V are real, while the space of g k is invariant under complex conjugation.

The Decomposition
Following the example of the moving kink, for each value of the parameter t we introduce the decomposition At this point ω k is arbitrary, although we define B ‡ = B † /(2ω k ).It is included to allow us to simplify expressions in applications.Note that as a result B ‡ and B do not necessarily satisfy an oscillator algebra, however their algebra may derived by inverting (6.9) and using the canonical commutation relations for ϕ(x) and π(x).We will argue in a moment that, although the operators B ‡ and B themselves implicitly depend on time, the resulting algebra is independent of time.
Using Eq. ( 6 Now clearly acting on a state with the operator O at time t = 0 is equivalent to acting on it with O (t) at time t, up to corrections of order O( √ λ).Therefore we may use the algebra of the operators O, that is to say the operators π 0 , ϕ 0 , ϕ t , B ‡ k and B k , to construct our comoving frame, one-soliton sector states at time t = 0.For example, we may impose that a state of interest is annihilated by some O 1 and the excited state is created by acting with O 2 .Then, at time t, our state will be annihilated by O (t) 1 and the excited state will be created by acting with O (t) 2 .This much is obvious, but then we may use the invertible expansions (6.13) to go between the O (t) operators and the fields ϕ(x) and π(x) at any time t, which allows us to use perturbation theory to study the static and dynamic properties of these states at any time.

Remarks
If the solution f (x, t) is periodic then the parameter t labeling the comoving frames is also periodic.Therefore, after one period, one returns to the original frame.If the original state was defined in the comoving frame by being annihilated by some set of operators O i , then after one period it will still be annihilated by those operators and so will still be in the same state.In particular, it will not have decayed.Thus, if such a state is sensible, for example if it does not posess classically unstable modes which would lead its perturbative expansion to be ill-defined, then it is stable at order O(λ 0 ), in other words over timescales of order O(1/m).In this case one would have disproved the claim that oscillons decay already at the linear level in Refs.[5,6].Indeed, this suggests more generally that the classical stability of a periodic classical solution generically implies an absence of decays via the emission of pairs of mesons.
So far we have only worked out one rather trivial example, the moving kink.However, with this formalism developed, we hope to turn to more interesting applications.We intend to approach quantum oscillons, estimating their lifetime, and also Rindler space, to see how quantum corrections affect the incoming radiation, trying to understand the remarkable yet perplexing results of Refs.[35,16,36,37].