A class of QFTs with higher derivative field equations leading to standard dispersion relation for the particle excitations
aa r X i v : . [ h e p - t h ] N ov A class of QFTs with higher derivative field equations leading tostandard dispersion relation for the particle excitations
T. PadmanabhanIUCAA, Pune University Campus,Ganeshkhind, Pune- 411 007. email: [email protected]
Abstract
Given any (Feynman) propagator which is Lorentz and translation invariant, it is possible to con-struct an action functional for a scalar field such that the quantum field theory, obtained by pathintegral quantization, leads to this propagator. In general, such a theory will involve derivatives ofthe field higher than two and can even involve derivatives of infinite order. The poles of the givenpropagator determine the dispersion relation for the excitations of this field. I show that it is possibleto construct field theories in which the dispersion relation is the same as that of standard Klein-Gordanfield, even though the Lagrangian contains derivatives of infinite order. I provide a concrete exampleof this situation starting from a propagator which incorporates the effects of the zero-point-length ofthe spacetime. I compare the path integral approach with an alternative, operator-based approach,and highlight the advantages of using the former.
I begin by describing how one can construct a QFT, if we are given a Lorentz and translation invariantpropagator. I will argue that this is indeed possible in the path integral formalism but there are someissues in the operator-based (canonical) quantization approach. Let us see how these results come about.It will be useful to keep the discussion somewhat general in order to understand various nuances.Consider a scalar field theory, which, when quantized by some procedure leads to a translationally invariant2-point function of the form: iG ( x , x ) ≡ iG ( x − x ) = Z d p (2 π ) e − ip ( x − x ) [ F ( p ) − iǫ ] (1)where F ( p ) is a real scalar function, in the momentum space, of its argument p ≡ p a p a , such that theintegral exists. I want to obtain this G ( x ), with x ≡ ( x − x ), as the propagator for a suitable scalarquantum field theory. This is completely straightforward in the path integral approach. Consider a Lorentzinvariant scalar field φ ( x ) with the 4-dimensional Fourier transform φ ( p ). I choose the action for the scalarfield to be: A ≡ − Z d p (2 π ) φ ∗ ( p ) F [( p )] φ ( p ) = − Z d x φ ( x ) F [( − (cid:3) )] φ ( x ) (2)(In the simple case of F ( p ) = F std ( p ) ≡ − p + m , the above action reduces to that of standard Klein-Gordon (KG) field.) It is trivial to perform a path integral quantization using the action in Eq. (2). The One motivation for this exercise is to investigate how the propagator with zero-point-length (see e.g., ref. [1]) can beobtained from quantizing a scalar field. But the question is of more general interest. Z ( J ) = Z D φ ( x ) exp (cid:20) iA [ φ ( x )] + i Z d x [ J ( x ) φ ( x )] (cid:21) (3)I assume that F in Eq. (2) is replaced by F − iǫ to ensure convergence. (This standard iǫ regulator adds aterm iǫφ to the Lagrangian.) Since the action is quadratic, it is straight-forward to evaluate this Gaussianpath integral and identify the propagator as the inverse of F . In momentum space this will lead to thepropagator function iG ( p ) = [ F ( p ) + iǫ ] − . In real space, it will lead to the propagator in Eq. (1), whichwhat I want. So, from the point of view of path integral quantization, it is easy to reverse-engineer anytranslation invariant propagator and obtain the action functional for a scalar field. I did not have to worryabout equal-time-commutation-relation (ETCR), canonical quantization etc. etc. thanks to the existenceof the path integral.
One can also do the same analysis in the Euclidean sector, thereby avoiding the iǫ prescription. Here p will be replaced by − p E (with p E being positive definite Euclidean variable) and the Euclidean actionwill be defined using F ( − p E ) as long as it is positive definite. The Euclidean propagator will be G E ( p E ) =1 /F ( − p E ) which, of course, gives the standard result for F std ( − p E ) = p E + m . The Euclidean approachis often easier to handle algebraically.The above procedure allows you to construct a QFT which, via path integral, can produce any trans-lation and Lorentz invariant propagator G ( x − y ). The simple steps in the construction are as follows:Start with a propagator G ( x − y ) of your choice (which depends only on ( x − y ) to ensure translation andLorentz invariance), find its Fourier transform G ( p ) and define F ( p ) as the reciprocal F ( p ) = 1 /G ( p );then the action in Eq. (2) with this F ( p ) will give rise to the propagator G ( x − y ) in the path integralapproach.Let me illustrate this procedure for a propagator which incorporates the zero-point-length in flat space-time [1]. In this case it is convenient to work in the Euclidean sector. In the Schwinger representation ofthe Euclidean propagator, this propagator is given by: G QG ( x ) = 116 π Z ∞ dss exp (cid:18) − sm − x + L s (cid:19) = 14 π m √ x + L K [ m p x + L ] (4)This involves replacing x by x + L in the Schwinger kernel. The momentum space propagator isobtained by a Fourier transform: G QG ( p ) = Z ∞ ds exp (cid:20) − s ( p + m ) − L s (cid:21) = L p p + m K [ L p p + m ] (5)Clearly, we get back the standard results (like e.g. G std ( p ) = ( p + m ) − ) when L = 0, as we should.Since K ( x ) is a monotonically decreasing, positive function of its argument with K ( x ) ≈ /x near x = 0,it is clear that the only pole G QG ( p ) occurs at p = − m , with unit residue. (More precisely, on analyticcontinuation to the Lorentzian sector with p E → − p , the pole is at p = m with unit residue.) To There is an extensive discussion of this propagator in the literature; some of the earlier papers are [1, 2] and for a sampleof more recent work, see [3]. In this work, I use G QG ( x ) purely as a non-trivial example to illustrate the ideas developedhere, without going into details of how G QG ( x ) is obtained etc. Those who are interested in more details of G QG (like e.g.,the motivation, construction, properties etc.), can find these in the cited references; in particular, some of these aspects aresuccinctly summarized in ref. [4]. G QG ( p ), we identify the function F in Eq. (2) as: F QG ( p ) = 1 G QG ( p ) = p p + m LK [ L p p + m ] (6)The QFT is now defined in terms of the action in Eq. (2) with this choice of F QG ( p ) in the momentumspace. The real space version in the in the Lorentzian sector is obtained by the standard replacements p E → − p → (cid:3) , and the addition of − iǫ regulator, leading to: F QG ( (cid:3) ) = √ (cid:3) + m LK [ L √ (cid:3) + m ] − iǫ (Lorentzian sector) (7)Obviously, the theory reduces to the Klein-Gordon limit, with F std ( (cid:3) ) = (cid:3) + m − iǫ , in the limit of L →
0. So we have succeeded in obtaining our propagator via path integral quantization of a scalar fieldwith a non-trivial action functional.As an aside let me comment on the heat kernel for the theory. We have the trivial identity: G ( p ) = Z ∞ ds e − sF ( p ) ; F ( p ) = 1 /G ( p ) (8)which, of course holds for any function G ( p ). The propagator in space(time), given by the Fouriertransform of G ( p ), can be expressed in the Schwinger proper time representation as: G ( x ) = Z d p (2 π ) e ip · ( x − x ) G ( p ) = Z ∞ ds h x | e − sF ( − (cid:3) ) | x i ≡ Z ∞ ds H ( x , x ; s ) (9)where x = x − x and the last equality defines the integrand H ( x , x ; s ). It is easy to verify that H satisfies the standard heat kernel equation and the boundary condition, ∂ H ∂s + F ( − (cid:3) ) H = 0 ; H ( x , x ; 0) = δ ( x − x ) (10)So H ( x , x ; s ) can indeed be thought of as the heat kernel for the QFT we have constructed. In momentumspace, this heat kernel is given by: H ( p ; s ) = exp − sF ( p ) = exp − " s p p + m LK [ L p p + m ] (11)On the other hand, we can read off from Eq. (5) another function ¯ H by writing: G QG ( p ) = Z ∞ ds exp (cid:20) − s ( p + m ) − L s (cid:21) ≡ Z ∞ ds ¯ H ( p ; s ); (12)with ¯ H ( p ; s ) = exp (cid:20) − s ( p + m ) − L s (cid:21) (13)Both ¯ H ( p ; s ) (in Eq. (13)) and H ( p ; s ) (in Eq. (11)) lead to the same propagator on integration over s . Ofthese two, H satisfies the standard heat kernel equation and the boundary condition while ¯ H does not. Sowhile the propagator is unique, its Schwinger propertime representation is not.3et us now consider the excitations and their dispersion relation in such quantum field theories. Ingeneral, the poles of the propagator G ( p ), or — equivalently — the zeros of F ( p ) determine the dispersionrelation. In the simplest case of Klein-Gordan theory, with F std ( p ) = − p + m , the dispersion relationis given by the standard one p = m . Of course, for a general F ( p ) the interpretation will not be easy.For example, if F ( p ) = ( − p + m )( − p − M ) we will have a tachyonic state with p = − M as well inthe model; if the poles are at complex values one has to describe them in terms of instabilities, decay etc.Remarkably enough, there is a class of theories for which these issues do not arise and the dispersionrelation is still given by the standard one, p = m even though the propagator is non-trivially different from the KG propagator. To identify this class of theories, I will again start with the propagator definedthrough Eq. (1) but now impose two conditions on the function F ( p ):(a) The only solution to F ( p ) = 0 — in the entire complex plane — is p = m where m is a real,Lorentz invariant, parameter with the dimensions of inverse length.(b) | F ′ ( m ) | = 1.This is same as demanding that the function 1 /F ( p ) has only one pole in the complex plane at p = m and the residue at the pole is unity. Obviously, the choice F std ( p ) = − p + m will satisfy both theseconditions and will lead to the standard QFT of a massive scalar field. It is also easy to see that thefunction F QG ( p ) in Eq. (6), for example, also satisfies these conditions. If two different functions — viz.,1 /F QG ( p ) and 1 /F std = 1 / ( − p + m ) — have identical poles and residues in the complex plane, thentheir difference will be an entire function f ( z ). If f ( z ) is bounded , then it has to be a constant. However,when f ( z ) is unbounded, we will get a theory which is non trivially different from the standard QFT. Thisis precisely which happens in the case of the function F QG ( p ) in Eq. (6). Also note that, while the integralin Eq. (1) exists for the choice of F QG ( p ) in Eq. (6), it cannot be evaluated by the usual procedure ofclosing the complex contour and picking up the residues; this is because F QG ( p ) in Eq. (6) differs from( − p + m − iǫ ) by an unbounded function in the complex plane.To summarize, QFTs with the action in Eq. (2) with F ( p ) satisfying the conditions (a) and (b)above have particle excitations which obey the standard KG dispersion relation. But these theories arestructurally very different from the KG scalar field theory and have a non-trivially different propagator1 /F ( p ). A simple, physically relevant, example is given by the propagator with zero-point-length and theassociated QFT with F ( p ) in Eq. (6). As a bonus, we also see that the addition of the zero-point-lengthby this procedure does not modify the dispersion relation for the excitations.Let us now ask what happens when we try to quantize the field φ in the action in Eq. (2) by the usualcanonical quantization procedure. The standard approach to canonical quantization involves identifyingthe canonical momentum π conjugate to the field φ and imposing the ETCR on φ and π . When F ( q )is not a linear function of q , i.e., when we go beyond the Klein-Gordon structure, you cannot write theaction in Eq. (2) with a quadratic kinetic energy term. This implies that the momentum conjugate to φ is no longer given by ∂ t φ . Therefore the standard procedure of imposing ETCR and then deriving thecommutators for creation and an annihilation operators etc. will not work.When F ( p ) is a polynomial (like e.g, a product of factors ( − p + m i ) with i = 1 , , ...N ) there isanother standard procedure to perform the canonical quantization by introducing a set of auxiliary fields,essentially to reduce the degree of derivatives appearing in the Lagrangian (see e.g., [5] and referencestherein.). In this case, G ( p ) will have N poles at m i with i = 1 , , ...N . But the case I am interestedin has a non-polynomial F ( p ) and — more crucially — the G ( p ) has only one pole at p = m ; see e.gthe F ( p ) in Eq. (6). This means the dispersion relation for excitations in the theory is the same as thatof the standard KG field.
If one attempts a canonical quantization with auxiliary fields, we will now need4 nfinite number of auxiliary fields. It is not clear how they will all combine together to give the standarddispersion relation for the excitations of the theory.There is another way to approach this issue, taking advantage of the fact F ( p ) satisfies the twoconditions (a) and (b) mentioned earlier. I will call this procedure canonical- like quantization since thereare some key differences. The action will now involve — for non-polynomial F , like the one in Eq. (6) weare interested in — infinite number of derivative terms if F is defined through a Taylor series expansion.The variation of the action in Eq. (2), in the momentum space, leads to the classical equations of motionfor the theory given by F ( p ) φ ( p ) = 0. This tells you that the solution φ ( p ) should have the structure φ ( p ) = δ D [ F ( p )] Q ( p ) (14)where Q ( p ) is an arbitrary function which is regular at p = m . The conditions (a) and (b) on the function F ( p ) allow us to simplify the Dirac delta function δ D [ F ( p )]. Using the standard rule for the Dirac deltafunction of the argument and the conditions (a) and (b) satisfied by the function F we immediately seethat the contribution from the only root gives δ D [ F ( p )] = δ D ( − p + m ). Therefore we can write thesolutions of the field equation as φ ( p ) = δ D [ F ( p )] Q ( p ) = δ D ( − p + m ) Q ( p ) (15) This tells you that the solutions to the field equation have the same on-shell structure as the solutions tothe standard Klein-Gordon equation which we obtain when F = F std = − p + m . So, a general solutionto the real scalar field can be expressed as a linear superposition φ ( x ) = Z d p (2 π ) ω p (cid:0) A p e − ipx + c.c (cid:1) ; ω p = p p + m (16)where px ≡ ω p t − p · x and A p is an arbitrary function. The structure on the right hand side is Lorentzinvariant and in fact is identical to what you start with in the case of Klein-Gordon equation. Let us now try to give a quantum meaning to the classical solution in Eq. (16). In the absence ofuseful canonical momentum and ETCR, we will use an alternative procedure: We start by promoting theexpansion in Eq. (16) directly into an operator-valued equation and then postulate the commutation rule[ A p , A † q ] = (2 ω p ) (2 π ) δ D ( p − q ) (17)This uses the Lorentz invariant structure for Dirac delta function; usually one works with the rescaledoperators a p ≡ p ω p A p so that [ a p , a † q ] = (2 π ) δ D ( p − q ). We can then introduce the number operator n k ≡ a † k a k with integer-labeled eigenkets and identify a Fock basis of kets |{ n k }i . That is, instead ofidentifying a canonical momentum from the Lagrangian and imposing ETCR, we are directly promotingthe solution to the field equation, given in Eq. (16) to operator-valued equation by introducing creation,annihilation operators and the Fock basis. This, in turn, will imply, purely algebraically, that the com-mutator of φ ( t, x ) and ∂ t φ ( t, y ) happens to be δ D ( x − y ); however, this is now just a consequence of ourpostulate in Eq. (17) and we do not identify ∂ t φ with some canonically conjugate momentum or imposeETCR between field and canonically conjugate momentum. In arriving at this result, I used the fact that the only zero of F ( p ) occurs in the real line. Any zero off the real line will,in general lead to mode functions (solutions) which are unbounded rather than oscillatory. So if we insist that the modesof the scalar field should not exhibit any instability, then the two conditions (a) and (b) which we imposed on F ( p ) aremandatory.
5o far, I have not introduced a Hamiltonian for the theory. This is, however, possible along the followinglines. We already know the field equation and its solution [given by Eq. (16)]. So we already know the complete time evolution of the operator φ ( t, x ) or — equivalently — the time evolution of creation andannihilation operators. From Eq. (16) we see that we can think of creation and annihilation operatorshaving the following time evolution: a k ( t ) = a k (0) e − iω k t ; a † k ( t ) = a † k (0) e iω k t (18)with the identification of a k appearing in the expansion Eq. (16) as a k (0). All I need to do is to identifya Hamiltonian operator which will induce this time evolution by standard Heisenberg equation of motion i∂ t a k ( t ) = [ a k , H ]. (This will ensure that i∂ t φ = [ φ, H ]). This is easy because we know from standardanalysis that, this will indeed be true if I postulate the Hamiltonian to be the sum: H = Z d k (2 π ) H k = Z d k (2 π ) ω k a † k a k (19)It immediately follows that the energy of the state |{ n k }i is given by the sum of E k = ω k n k . So thephysical states, the excitations spectra and the time evolution remains identical to the standard Klein-Gordon theory. Let me summarize the logical structure of what I have done so far, vis-a-vis canonical-likequantization:(i) I start with a well-defined classical field theory, generically involving derivatives higher that secondorder, based on a function F which satisfies two conditions (a) and (b). From the action functional inEq. (2) I obtain the field equation and a solution given in Eq. (16).(ii) To quantize the theory, I upgrade Eq. (16) as an operator equation and postulate the commutationrules in Eq. (17). This replaces the postulate of ETCR in the standard approach.(iii) Since I already know the time evolution of the field operator φ , I determine the Hamiltonian fromthe condition that standard Heisenberg equations of motion, i∂ t φ = [ φ, H ] should hold. This gives me theHamiltonian for the theory in Eq. (19).(iv) I do not use the classical action function to identify the conjugate momentum or to derive the formof the Hamiltonian. The conjugate momentum is needed only for ETCR and I replace the postulate ofETCR by the postulate in Eq. (17). The Hamiltonian is then derived/defined such that the time evolutionis consistent with the operator-valued field equation.These results should be — at least mildly — surprising. They show that one can have a very general fieldequation (with an arbitrary function F ) for which the excitation spectrum is the same as standard Klein-Gordon equation, representing spinless bosonic particles obeying the relativistic relation between energyand momentum. More importantly, this canonical-like quantization procedure, will lead to a completelydifferent propagator compared to what we found in the case of path integral. This arises as follows: We nowhave a well defined vacuum state and field operator; so one could define the propagator as the expectationvalue of the time ordered product of the field operators. This will lead, on using Eq. (16), to the standardresult: G ( x , x ) = h | T [ φ ( x ) φ ( x )] | i = Z d k (2 π ) ω k e − iω k | t | + i k · x ; x = x − x (20)Incidentally, if one determines the dispersion relation of the theory by looking at the poles of the propagator,we get the same, standard dispersion relation for all propagators of the form in Eq. (1), because of ourassumptions (a) and (b) about the function F ( p ). So even though the propagators obtained by the twomethods are widely different, the dispersion relation obtained from their poles remain the same.6t is rather interesting that the path integral actually picks up the propagator we started with, inEq. (1), at one go while it is unclear how to obtain it by canonical quantization procedure, after introducinginfinite number of auxiliary fields. If we take that approach we will have infinite pairs of propagators which,somehow, should combine together to give Eq. (1). This connection is worth investigating. To summarize, the key results of this work are the following: • Given any Lorentz and translation invariant propagator in Eq. (1), it is always possible to constructa scalar field action [see Eq. (2)] such that path integral quantization of the theory leads to thepropagator in Eq. (1). A non-trivial example is given by the propagator with zero-point-length (seeEq. (5)), which can be obtained from a scalar field theory with the choice of operator in Eq. (7). • For a general propagator of the form in Eq. (1), the excitation spectra and the dispersion relationcan be complicated and even ill-defined. However, there exists a subset of these models in which F ( p ) satisfies two conditions [listed as (a) and (b) above] for which a simple situation arises; inthese models, the excitation spectra and the dispersion relation are identical to those in standardKG theory. • The propagator with zero-point-length belongs to this subset of models. So we conclude that theaddition of zero-point-length does not change either the nature of particle excitations or modify thedispersion relations of the theory. • It is also possible to attempt an operator-based quantization of the action in Eq. (2) when F ( p )satisfies two conditions [listed as (a) and (b) above]. This leads to a structure very similar to that ofKG theory as far as the Fock basis is concerned. The propagator, defined as time ordered correlator,however, will not match with the one obtained from the path integral quantization; nevertheless, thedispersion relations are identical. Acknowledgement
I thank S. Date, A.D. Patel and S. Solodukhin for useful discussions. My research is partially supportedby the J.C.Bose Fellowship of Department of Science and Technology, Government of India.
References [1] T. Padmanabhan,
Phys. Rev. Letts , , 1854 (1997) [hep-th-9608182];T. Padmanabhan, Phys. Rev. , D 57 , 6206 (1998)[2] B. S. DeWitt, (1964)
Phys. Rev. Lett. , 114;T.Padmanabhan, (1985), Gen. Rel. Grav., , 215;T.Padmanabhan, (1985), Ann. Phys. , , 38;T.Padmanabhan, (1987), Class. Quan. Grav. , , L107; More generally, it is not clear whether this approach — canonical-like quantisation, with a rather ad hoc identificationof Hamiltonian etc. — leads to a completely consistent quantum theory. This requires further study.
7. Shankaranarayanan and T. Padmanabhan,
Int. Jour. Mod. Phys , D 10 , 351 (2001) [gr-qc-0003058];K.Srinivasan, L.Sriramkumar and T. Padmanabhan,
Phys. Rev.
D 58 , 044009 (1998) [gr-qc-9710104]Michele Fontanini, Euro Spallucci, T. Padmanabhan,
Phys.Lett.,
B633 , 627-630 (2006)[hep-th/0509090][3] Dawood Kothawala, L. Sriramkumar, S. Shankaranarayanan, T. Padmanabhan,
Phys.Rev.,
D 79 ,104020 (2009) [arXiv:0904.3217]Kothawala D and Padmanabhan T (2014),
Phys. Rev.
D 90
Phys. Rev.
D 88
Entropy , , 7420 [arXiv:1508.06286];D. Jaffino Stargen, D. Kothawala, (2015), Phys.Rev.
D 92
Symmetry (1) (2020) 138;[4] T.Padmanabhan, Phys. Letts. , B 809 (2020) 135774T. Padmanabhan,
J. High Energ. Phys. , 2020, (2020) [arXiv:2006.06701].[5] See e.g., G.W. Gibbons et al., Phys. Rev.