From classical Lagrangians to Hamilton operators in the Standard-Model Extension
In this article we investigate whether a theory based on a classical Lagrangian for the minimal Standard-Model Extension (SME) can be quantized such that the result is equal to the corresponding low-energy Hamilton operator obtained from the field-theory description. This analysis is carried out for the whole collection of minimal Lagrangians found in the literature. The upshot is that first quantization can be performed consistently. The unexpected observation is made that at first order in Lorentz violation and at second order in the velocity the Lagrangians are related to the Hamilton functions by a simple transformation. Under mild assumptions, it is shown that this holds universally. This result is used successfully to obtain classical Lagrangians for two complicated sectors of the minimal SME that have not been considered in the literature so far. Therefore, it will not be an obstacle anymore to derive such Lagrangians even for involved sets of coefficients — at least to the level of approximation stated above.
pacs:11.30.Cp, 45.20.Jj, 45.50.-j, 03.65.-w
Fundamental physics of the 21st century will be governed by the search for a theory of quantum gravity. This will ultimately bring the field of CPT- and Lorentz violation more into the focus of high-energy physics. One of the basic and most essential results obtained in this context is that Lorentz violation can arise naturally in closed-string field theory Kostelecky:1988zi (); Kostelecky:1991ak (); Kostelecky:1994rn (). Besides that, a violation of Lorentz invariance was shown to occur in other realms of high-energy physics or alternative approaches to quantum gravity such as loop quantum gravity Gambini:1998it (); Bojowald:2004bb (), theories of noncommutative spacetimes AmelinoCamelia:1999pm (); Carroll:2001ws (), spacetime foam models Klinkhamer:2003ec (); Bernadotte:2006ya (); Hossenfelder:2014hha (), quantum field theories in backgrounds with nontrivial topologies Klinkhamer:1998fa (); Klinkhamer:1999zh (), and last but not least, Hořava-Lifshitz gravity Horava:2009uw ().
These results have far-reaching consequences. First, they show that violations of the aforementioned fundamental symmetries may be signals of physics at the Planck scale. Second, Lorentz invariance is the very base of the established theories, i.e., the Standard Model and General Relativity. Hence, even if quantum gravity looks totally different from what physicists currently imagine these theories should be subject to precise experimental tests. This is only possible based on a framework that extends both the Standard Model and General Relativity to tell the experimentalists what effects they may expect to measure. The most general framework is the Standard-Model Extension (SME) Colladay:1998fq (). The latter is an effective field theory based on the Standard Model, which comprises all Lorentz-violating operators of mass dimension 3 and 4 that can be added to the Lagrange density. The nonminimal SME Kostelecky:2009zp (); Kostelecky:2011gq (); Kostelecky:2013rta () additionally includes all higher-dimensional operators. Whenever Lorentz violation is studied, CPT violation is automatically taken into account due to a theorem by Greenberg Greenberg:2002uu (), which says that in effective field theory CPT-violation implies Lorentz noninvariance.
Various theoretical investigations of the SME have been carried out at tree-level Kostelecky:2000mm (); oai:arXiv.org:hep-ph/0101087 (); Casana:2009xs (); Casana:2010nd (); Klinkhamer:2010zs (); Schreck:2011ai (); Casana:2011fe (); Hohensee:2012dt (); Cambiaso:2012vb (); Schreck:2013gma (); Schreck:2013kja (); Schreck:2014qka (); Colladay:2014dua (); Casana:2014cqa (); Albayrak:2015ewa () and higher-orders in perturbation theory Jackiw:1999yp (); Chung:1999pt (); PerezVictoria:1999uh (); PerezVictoria:2001ej (); Kostelecky:2001jc (); Altschul:2003ce (); Altschul:2004gs (); Colladay:2006rk (); Colladay:2007aj (); Colladay:2009rb (); Gomes:2009ch (); Ferrero:2011yu (); Casana:2013nfx (); Scarpelli:2013eya (); Cambiaso:2014eba (); Santos:2014lfa (); Santos:2015koa (); Borges:2016uwl (); Belich:2016pzc (). These results are important as they demonstrate that the SME is a viable framework to investigate Lorentz violation. Therefore, experimental studies are warranted as well where a great deal of sharp experimental bounds on the minimal SME already exist opening the pathway to covering the leading-order operators of the nonminimal SME. A yearly updated compilation of all constraints can be found in Kostelecky:2008ts ().
After setting up the particle-physics part of the minimal SME in Colladay:1998fq () the gravity part was established in Kostelecky:2003fs (). One of the most crucial results of the latter paper is that explicit Lorentz violation is incompatible with gravity where the incompatibility is due to the Bianchi identities of Riemannian geometry. Therefore, in a curved background Lorentz violation can only be studied consistently when the symmetry is broken spontaneously, cf. Kostelecky:1989jp (); Kostelecky:1989jw (); Bluhm:2008yt (); Hernaski:2014jsa (); Bluhm:2014oua (); Maluf:2014dpa (). An alternative possibility of circumventing the incompatibility could be to use a geometric concept other than Riemannian geometry. Because of that reason, Lorentz violation based on Finsler geometry Finsler:1918 (); Cartan:1933 (); Bao:2000 (); Antonelli:1993 () is currently being investigated extensively. Finsler geometry can be regarded as Riemannian geometry without the quadratic restriction of line intervals, i.e., any possible interval obeying certain reasonable properties can be considered. Line intervals can also involve preferred directions on the manifold, which is why this geometric approach is interesting for people studying Lorentz violation.
To consider Finsler geometry in the SME a reasonable starting point is needed. General Relativity and possible extensions of it reside in the realm of classical physics. However, the particle-physics part of the SME is a field theory concept, i.e., one has to map the field theory description to a classical-physics analog. This was carried out for various cases of the minimal SME Kostelecky:2010hs (); Kostelecky:2011qz (); Kostelecky:2012ac (); Colladay:2012rv (); Schreck:2014ama (); Russell:2015gwa () and a number of nonminimal cases Schreck:2014hga (); Schreck:2015seb (). The results of these studies are Lagrangians for classical, relativistic, pointlike particles including Lorentz violation based on the SME. Further analyses of these Lagrangians or investigations in other sectors of the SME can be found in Silva:2013xba (); Foster:2015yta (); Colladay:2015wra (); Schreck:2015dsa (); Silva:2015ptj (). The classical Lagrangians obtained were shown to be related to Finsler structures Kostelecky:2011qz (); Kostelecky:2012ac (); Colladay:2012rv (); Schreck:2014hga (); Schreck:2015seb () and can possibly serve to study explicit Lorentz violation in curved backgrounds, cf. Kostelecky:2011qz (); Schreck:2015dsa ().
The mapping investigated in the papers mentioned above starts at the quantum description of the SME and it ends at the classical regime. Therefore, the motivation of the current article is to answer one question. Assuming that we have the classical Lagrangians only and do not know about the field theory description of the SME, is it possible to quantize the classical theory to arrive at the quantum-mechanical Hamiltonian based on the SME? Note that the SME is a relativistic field theory, which allows for obtaining the corresponding low-energy Hamiltonian with the Foldy-Wouthuysen procedure Foldy:1949wa (). Hence, an alternative method could be to expand the relativistic classical Lagrangians in the ratio of the velocity and the speed of light and to perform quantization subsequently. Finding out whether or not this method works is the goal of the paper. In the course of the analysis, the unexpected result is encountered that the classical Lagrangians and Hamilton functions considered are related by a simple transformation at first order in Lorentz violation and at second order in the momenta. This observation may be interesting and important in practice.
The paper is organized as follows. In Sec. II we compile all Lagrangians obtained in the literature so far. Thereby, the velocity-momentum correspondence and the classical Hamilton function is computed for each. Section III is dedicated to the first quantization of the results. All classical momenta are promoted to quantum operators and a suitable Ansatz for a spin structure is introduced. It is shown that the quantum-mechanical Hamilton operators can be obtained consistently. In Sec. IV the leading-order expansion of each Lagrangian is investigated more closely. By doing so, we find the simple relation between the Lagrangians and the Hamilton functions mentioned above. Under mild assumptions, it is shown that this result is valid in general. Subsequently, we apply it to two complicated cases of the minimal fermion sector not considered in the literature so far. Last but not least, all findings are discussed and concluded on in Sec. V. Throughout the paper, natural units with are used unless otherwise stated.
Ii Classical Hamilton functions
In Kostelecky:2010hs () the procedure was set up to assign a classical Lagrangian to a particular case of the SME fermion sector. Consider a quantum wave packet that is a superposition of plane-wave solutions to the free-field equations with a suitable smearing function. If the smearing function in configuration space is chosen to fall off sufficiently fast outside of a localized region this wave packet is interpreted as a particle in the classical limit. The physical propagation velocity of the packet corresponds to its group velocity for most cases Brillouin:1960 (). Denoting the four-momentum of a plane wave, which is part of the wave packet, by and the four-velocity of the classical particle by we have the following five equations that govern the mapping procedure:
The first is the dispersion relation of the particular SME fermion sector considered. The second says that the group velocity of the quantum wave packet shall correspond to the three-velocity of the classical particle. These are three equations, one for each component. The last follows from the reasonable assumption of an action that is invariant under changes of parameterization. For exhaustive discussions on that procedure we refer to Kostelecky:2010hs ().
Within the current paper a classical Lagrangian shall not be obtained but it is supposed to be the starting point. The aim is to derive the quantum-mechanical Hamiltonian from this Lagrangian. To do so, the first step is to obtain the particle energy as a function of velocity for which there are two possibilities. In proper-time parameterization, and where is the three-velocity of the particle. According to we obtain
The second method is to derive the energy via a Legendre transformation, cf. Eq. (2.1c):
where the spatial momentum is understood to have upper indices. In what follows, both procedures are checked to lead to the same result, which is the particle energy as a function of the three-velocity. For quantization, the Hamiltonian has to be computed from the classical energy. To do so the energy is needed as a function of spatial momentum instead of velocity . Hence, it is necessary to solve
with respect to to give replacement rules for the velocity in favor of the momentum. The result is the classical Hamilton function , which forms the basis for quantization. In the forthcoming subsections, this procedure will be carried out for all classical Lagrangians found for the minimal SME fermion sector. We will work at first order in Lorentz violation and at second order in the momentum or velocity.
For each case of the SME fermion sector there are distinct Lagrangians for the particle and the antiparticle solutions. They are related to each other by the replacement . In this article we will only consider the particle Lagrangians as they deliver positive energies. Classically, there are no antiparticles after all. For the properties of the SME fermion coefficients we refer to Table 1 in Kostelecky:2013rta ().
We start with the Lagrangian for the coefficients that are of mass dimension 1. It is based on the observer four-vector with . The Lagrange function can be extracted from Eq. (8) or Eq. (12) in Kostelecky:2010hs () when setting all the other controlling coefficients to zero. It is comprised of the standard square root term and an observer Lorentz scalar involving both the four-velocity and the observer four-vector :
The energy as a function of velocity reads
where the latter result is exact in Lorentz violation and valid at second order in the velocity. The momentum is then given by
This is solved with respect to the velocity,
and it is inserted into Eq. (2.6) to give the Hamilton function
Note that a term comprising the scalar product has emerged where there is no equivalent to such a term in Eq. (2.6). This demonstrates that it is crucial to keep track of the contributions at first order in Lorentz violation in the velocity-momentum correspondence of Eq. (2.8).
The Lagrange function associated to the dimensionless observer four-tensor coefficients shall be considered next where we define . The exact result can be found in Eq. (10) of Kostelecky:2010hs () and it involves both the symmetric and the antisymmetric part of . However, the antisymmetric part contributes at second order in Lorentz violation only. Since in this article all considerations are restricted to first order in Lorentz violation we start with the first-order expansion of the latter Lagrange function, which is given by
Here only the symmetric part of contributes as expected. The particle energy as a function of the velocity then reads as follows:
Note that the latter does not involve the mixed components with a timelike and a spacelike index. This is different for the equations relating the spatial momentum to the velocity:
Therefore, replacing the velocity by the momentum in Eq. (II.2) introduces coefficients into the Hamilton function:
The next case to be studied is the observer four-vector with including dimensionless coefficients. The Lagrangian follows from Eq. (8) in Kostelecky:2010hs () by setting the and coefficients to zero:
where its structure is very similar to the Lagrangian of given in Eq. (2.5). The particle energy as a function of velocity reads
It does not involve the spatial components of , which is a behavior similar to Eq. (2.6) that does not comprise the spatial components of either. The correspondences between velocity and momentum read
Replacing the velocity by the momentum in Eq. (2.15) introduces the spatial components of into the energy, which works in analogy to the case of the coefficients:
These results demonstrate that the families of the minimal and coefficients behave in a very similar manner. This is not surprising since the and coefficients are part of the same effective coefficient, cf. the first of Eq. (27) in Kostelecky:2013rta ().
The three frameworks previously considered do not break degeneracy with respect to the particle spin, i.e., there is only a single classical Lagrangian corresponding to the particle solutions in quantum field theory. For the following cases this degeneracy is broken, starting with the coefficients of mass dimension 1. The Lagrangian is obtained from Eq. (12) in Kostelecky:2010hs () by setting the coefficients to zero:
The two signs break the spin degeneracy as mentioned. The upper sign always corresponds to the configuration of “spin-up” and the lower to “spin-down.” Although the concept of spin does not exist classically this correspondence can be inferred from the quantum theory. We will come back to this point later. Note that the Lorentz-violating contribution has a very different structure compared to the cases of the , , and coefficients; it is called bipartite Kostelecky:2012ac (). The Lagrangian was shown to be related to a Finsler space that is neither Riemannian nor of Randers-type Kostelecky:2010hs (). One peculiarity is that it is not straightforward to expand Eq. (2.18) with respect to the controlling coefficients or the velocity. Therefore, we consider two observer frames: the first with being purely spacelike and the second with purely timelike.
The first case is based on a purely spacelike observer four-vector that is expressed as with the spatial part where the latter will occur in all results. For such a choice the Lagrangian of Eq. (2.18) reads as follows:
The energy can then be obtained just as before by differentiating the Lagrangian with respect to or by a Legendre transformation. The result shall be expanded at first order in Lorentz violation. This is a bit more involved compared to the previous cases since the Lorentz-violating contribution behaves asymptotically like a square root function that does not have a Taylor expansion for a vanishing argument. This can be remedied by introducing the angle between the velocity and the three-vector composed of the controlling coefficients. After doing so, the magnitude of can be extracted from the expression which allows for a subsequent expansion with respect to the velocity:
We apply the same method to perform expansions of the momentum given as a function of the velocity:
The latter is then solved for the velocity components to give
At this point there is a subtle issue that occurs for the first time in the course of our studies. When solving Eq. (2.21) for the velocity we encounter an absolute value of the trigonometric function in Eq. (2.22). Eliminating these absolute-value bars would lead to four different sign choices dependent on both the angle and the upper or lower sign coming from the original Lagrangian. In the end this would result in four different Hamilton functions, which does not match the number of degrees of freedom in the original Lagrangian. Therefore, we have to set up a proposal telling us how to choose the signs appropriately to obtain two distinct Hamilton functions corresponding to the two Lagrangians given initially.
On the base of observer Lorentz invariance, a coordinate system is defined such that its axis points along the preferred direction . The spin quantization axis can be chosen freely and for convenience it is arranged to point along the axis as well, cf. Fig. 1. One the one hand, the spin-up state can then be understood to be realized in the upper half-plane. That corresponds to for the angle between the particle velocity and the preferred axis. On the other hand, the spin-down state is realized in the lower half-plane where . Now assume that the particle is in a spin-up state. For the absolute value bars around do not have any effect, which is why they will be dropped, choosing the upper of the two signs in Eq. (2.22). For the absolute value bars act like a minus sign before the term affected. Therefore, upon dropping them, the lower sign is picked. So each sign is not valid for all momentum configurations possible but just for a restricted range of angles. For the particle being in a spin-down state, the same procedure can be applied with both signs switched. Then for the minus sign must be chosen and for the plus sign.
Finally, in Eq. (II.4.1) the velocity is replaced by the momentum leading to the Hamilton function:
In the last step we introduced the cross product between the spatial momentum and the spacelike vector of the controlling coefficients. Note that in general the velocity vector does not point along the direction of the momentum vector, cf. Eqs. (2.21), (2.22). Hence, the angle between and deviates from the angle between and . However, the deviation is of first order in Lorentz violation (see, e.g., Eqs. (13), (16) in Kostelecky:2010hs () for the and coefficients, respectively), which leads to second-order corrections in Eq. (II.4.1) that are discarded anyhow. With the spin quantization axis pointing along the axis and the particle being in a spin-up state, the upper sign of the Hamilton function must be picked for and the lower sign for . When the particle is in a spin-down state, both signs have to be switched.
To study the second case of the coefficients an observer frame is chosen where is purely timelike. Such a choice involves a single controlling coefficient: . The Lagrangian is isotropic and takes a simple form:
Since the Lorentz-violating contribution does not comprise the energy corresponds to the standard expression when expressed in terms of the velocity:
However, this is not the case for the momentum since the latter is obtained as the first derivative of the Lagrangian with respect to . In proper-time parameterization we obtain
Solving this relation for the velocity leads to
Note the singularities in and in the latter two expressions. Furthermore, here it must be distinguished between the two signs: the upper sign has to be chosen for and the lower for . This procedure differs from the prescription that we introduced in the last section. It is challenging to illustrate it physically by taking into account the particle spin just as we did in Sec. II.4.1. It seems that a similar procedure always has to be carried out when there are singularities in , in the momentum-velocity correspondences, cf. the forthcoming Secs. II.5.2, II.6.3. Now, replacing by in Eq. (2.25) introduces the single controlling coefficient into the Hamilton function:
The resulting expression is isotropic, as expected, and does not have any singularities.
The next cases to be studied involve the observer two-tensor coefficients that are of mass dimension 1. Several Lagrangians valid for particular subsets of coefficients were obtained in the literature. In this context the following observer Lorentz scalars are helpful:
where with is the totally antisymmetric Levi-Civita symbol in four spacetime dimensions. The matrix can be taken as antisymmetric, cf. Kostelecky:2000mm ().
Spacelike case with and
The first Lagrangian considered is valid for and . It is given by Eq. (15) in Kostelecky:2010hs ():
Spin degeneracy is again broken just as for the coefficients. An important choice that fulfills and is an antisymmetric comprised solely of controlling coefficients with spacelike indices:
The energy bears some similarities to Eq. (II.4.1). Expansions with respect to the controlling coefficients and the velocity are computed as before. We introduce the angle between the velocity vector and the vector comprising the controlling coefficients. Then is extracted from the square root and the resulting expression is expanded with respect to the velocity:
Similarly, this procedure is applied to obtain the momentum:
which is then solved for the velocity:
Here the same issue appears that we encountered for the spacelike case of the coefficients in Sec. II.4.1, i.e., the sign choice depends on the angle . When the spin quantization axis points along the direction and the particle is in a spin-up state, we choose the upper sign for and the lower for . The signs must be picked vice versa for the particle being in a spin-down state. Plugging the velocity-momentum correspondence of Eq. (2.34) into Eq. (II.5.1), the final result is the Hamilton function:
In the last step we expressed the result by the scalar product of and in analogy to how we dealt with Eq. (II.4.1) by introducing the cross product. Note that the angle does not correspond to the angle between and . However, deviations are of first order in Lorentz violation, which is why those produce higher-order terms in the final result of Eq. (II.5.1). For spin pointing up, the upper sign of the Hamilton function holds for and the lower sign for , cf. Sec. II.4.1. For spin pointing down, the opposite is the case.
Timelike case with and
Another case with is constructed from an antisymmetric with nonzero coefficients having one timelike index. This choice reads as follows:
That sector is also based on the Lagrangian in Eq. (2.30). The energy is obtained from its first derivative with respect to :
and it does not involve any Lorentz-violating terms at first order. This is different for the momentum and the velocity that are given by:
Both expressions are singular in and , respectively. Besides, the upper sign holds for and the lower for . Recall that the timelike case of the coefficients behaved similarly, cf. Sec. II.4.2. The singularities do not occur in the Hamilton function, though:
Just as before, we introduce the cross product between and the three-momentum neglecting higher-order terms in Lorentz violation.
Case with and
The case of with and was considered in Kostelecky:2010hs () as well. This particular framework is much more complicated than the previous one, which becomes manifest in the polynomial of their Eq. (18) whose (perturbative) zeros with respect to correspond to the Lagrange functions searched for. In Kostelecky:2010hs () they are not stated explicitly due to their complicated structure. Here we consider the following configuration of nonvanishing controlling coefficients:
Evidently, can be expressed by a timelike preferred direction and a spacelike one , i.e., these directions are physical and especially the spacelike one will appear in the results below. For this choice the Lagrange functions are obtained from Eq. (18) of Kostelecky:2010hs () with computer algebra where a subsequent expansion of the result at first order in Lorentz violation leads to
The Lorentz-violating contribution can be expressed by the observer Lorentz scalar . This means that the form of the result stays unchanged in an arbitrary observer frame that can be transformed to by an observer Lorentz transformation. At first order in Lorentz violation, the Lagrangian corresponds to Eq. (2.30) for . The energy is then given by
which can be written in terms of the spatial part of the second preferred direction, cf. Eq. (2.40b). The momentum-velocity correspondence reads
where . From this we obtain the Hamilton function:
Note the similar structure of the result in comparison to Eq. (II.5.3). As mentioned above, the Lagrangian of Eq. (2.41) can be transformed to another observer frame. The special given in Eq. (2.40b) points along the first spatial direction of the coordinate frame. The general case results from an observer rotation such that points along an arbitrary direction.
The coefficients are dimensionless and comprised by an observer two-tensor that can be taken as traceless, cf. Kostelecky:2000mm (). The corresponding Lagrangian is challenging to be obtained since the dispersion relation is quartic in . Nevertheless, two cases were considered in Colladay:2012rv (). For convenience, we define a couple of helpful observer Lorentz scalars as follows:
Nonsymmetric operator with nonvanishing timelike components
The first case involves the nonvanishing coefficients only, i.e., the whole tensor is not symmetric. We introduce the three-vector including these coefficients, which turns out to be a useful quantity:
The corresponding Lagrangian is given by Eqs. (26), (27) in Colladay:2012rv (). Its form is rather complicated; it comprises both a cross product and a scalar product of and . To consider the expression at leading order in Lorentz violation and at second order in the velocity we follow the procedure already used for the and coefficients. Let be the angle between and the velocity. The Lagrangian then reads as follows:
with the velocity unit vector . The Lagrangian was constructed directly in proper-time parameterization, which is why it does not depend on . Therefore, the energy cannot be computed via Eq. (2.2) but we have to perform a Legendre transformation according to Eq. (2.3). The result at first order in Lorentz violation is given by:
The momentum reads
where the final expression is solved for the velocity:
The Hamilton function can then be computed as
This corresponds to Eq. (23) of Colladay:2012rv () at first order in Lorentz violation.
Antisymmetric operator with nonvanishing timelike components
For the second case considered in Colladay:2012rv (), is assumed to be antisymmetric. Furthermore, the quantity shall vanish. An important case that obeys these properties is an antisymmetric two-tensor with nonvanishing components only in the first row and column, respectively. Hence, the coefficients are taken to be nonvanishing again where . Additionally, we introduce the same vector as before: