How is Lorentz Invariance encoded in the Hamiltonian?
aa r X i v : . [ g r- q c ] J u l How is Lorentz Invariance encoded in the Hamiltonian?
Nirmalya Kajuri ∗ Department of Physics, Indian Institute of Technology Madras,Chennai 600036, India
Abstract
One of the disadvantages of the Hamiltonian formulation is that Lorentz invariance is not man-ifest in the former. Given a Hamiltonian, there is no simple way to check whether it is relativisticor not. One would either have to solve for the equations of motion or calculate the Poisson Brack-ets of the Noether charges to perform such a check. In this paper we show that, for a class ofHamiltonians, it is possible to check Lorentz invariance directly from the Hamiltonian. Our workis particularly useful for theories where the other methods may not be readily available. ∗ Electronic address: [email protected] . INTRODUCTION Consider the following Hamiltonian: H = 12 ( π A + ( ~ ∇ A ) + m A A + π B + ( ~ ∇ B ) + m B B ) + g ~ ∇ A ) B + 12 π A (cid:18) g B g B (cid:19) (1)where A , B are scalar fields and g is a coupling constant. Is Lorentz invariance a symmetryof the system described by the above Hamiltonian? The answer is yes. The correspondingLagrangian is given by: L = 12 (( ∂ µ A ) − m A A + ( ∂ µ B ) − m B B ) − g ∂ µ A ) B (2)Which is manifestly Lorentz invariant. But this was not at all obvious from the Hamil-tonian. Given a Hamiltonian how do we check for Lorentz invariance?Generally, one has three avenues : One, solve for the equations of motion and see if theseare relativistic.Two, calculate the Poisson brackets of the Noether charges of Lorentz symmetry with eachother and check if this gives a representation of the Lorentz algebra. We do not elaborateon this method here, the reader is referred to [1] for details. The non-trivial part of thismethod involves calculating the Poisson Bracket relations of the generators of boost andangular momenta with the Hamiltonian.Finally, one may construct the Lagrangian and check if it is Lorentz invariant. If one isgiven a quantum Hamiltonian, then this last method would involve obtaining a path integralrepresentation.In this paper we will show that for a class of physical systems, it is possible to obtain thecondition for Lorentz invariance directly from a Hamiltonian. As we will see, the conditionstates that certain vectors and tensors that may be constructed from the Hamiltonian areproportional to one another. We note that this method works when the representationof Lorentz symmetry is linear. There are cases where Lorentz symmetry is non-linearlyrealized, for instance gauge-fixed or reduced phase space treatments of gauge theories (seefor instance [2]), but these are not considered here.The systems for which the result holds must satisfy certain criteria. These are: where2he field momentum or the space derivative of one of the fields occurs in the Lagrangianthey must(i) Come in even powers . That is, any of the terms in the Lagrangian may contain aterm like ( ∂ x A ) but a factor like ( ∂ µ A ) C µ , where C µ may be a (pseudo)vector, cannot occurin any of the terms.(ii) Not be multiplied with the derivatives of one of the other fields. That is a term like( ∂ x A ) B is allowed but ( ∂ x A ) ( ∂ x B ) is not.This work is particularly important in the context of those quantum theories where theother avenues may not be readily available. One such case was polymer quantized scalarfield theory [3]. The status of Lorentz invariance of this theory was open till a path integralformulation was found recently [4], establishing that the theory is not Lorentz invariant.However our work makes the deduction of Lorentz non-invariance of this theory extremelysimple, as we will show.The plan of the paper is as follows. In the next section we start from the Lagrangianformulation and show how to derive the condition for Lorentz invariance for these Hamil-tonians. First we restrict ourselves to an even more limited (but still non-trivial) class ofHamiltonians, for which the condition for Lorentz invariance takes a particularly simpleform. Then after presenting some examples we sketch how to extend the derivation for allHamiltonians satisfying (i) and (ii) above. The final section summarizes our findings. II. ENCODING RELATIVISTIC INVARIANCE IN THE HAMILTONIAN: THERESTRICTED CASE
In this section we derive a sufficient condition for Lorentz invariance for physical systemswhich satisfy (i) and (ii) above. First we will present the derivation for systems which satisfyanother criteria :(iii) The derivative terms appearing in the Lagrangian must be quadratic. That is, any ofthe terms in the Lagrangian may contain the factor ( ∂ x A ) but a factor like ( ∂ µ A ) cannotoccur in any of the terms.Note that the system described by (2) (equivalently (1)) satisfies these criteria.For all such Hamiltonians, the condition for Lorentz invariance takes a particularly simple3orm. It goes as: construct for each field A, the following two column matrices - F µA = ( ∂H∂π A , ~ ∇ A ) (3) G µA = ( π A , ∂H∂ ~ ∇ A ) (4)The condition for Lorentz invariance is, for each A : F µA = kG µA (5)where k is a scalar which may be the function of fields and its derivatives.In the following subsection we derive (5) starting from a Lagrangian formulation. Thiswill be followed by some examples. We will complete this section by sketching the steps ofthe derivation for a system satisfying only (i) and (ii). A. Derivation from the Lagrangian
In this section we will demonstrate how, for systems satisfying criteria (i) and (ii), theLorentz invariance of a Lagrangian is encoded in the Hamiltonian through (5). We’ll exhibitthe proof for a system of scalar fields. The extension to higher spin fields is straightforward.Let us consider a relativistic Lagrangian L which is a function of scalar fields φ n . Wewrite this as: L = L + L where all terms containing derivatives in the field are put in L and the rest of the termsare in L . For instance, in the Lagrangian (2) we will have L = 12 (( ∂ µ A ) + ( ∂ µ B ) ) + g ∂ µ A ) B L = − m A A − m B B (6)Now since L does not contain any terms containing field derivatives, it follows that L and L must be separately Lorentz invariant. Now for a Lagrangian satisfying the criteria(i) and (ii), L may be written as: L = X n ∂L ∂ ( ∂ µ φ n ) ∂ µ φ n (7)4gain, each term in L must be Lorentz invariant in itself. But from the above, each termin L can be written in the form ( L ) n = 12 η µν W µn Z νn (8)where W µn = ∂ µ φ n (9) Z νn = ∂L ∂ ( ∂ µ φ n ) (10)This is manifestly a Lorentz invariant quantity if W µn and Z n ν transform as four vec-tors under Lorentz transformations. But W µn are a Lorentz four vectors by definition. Ittherefore follows that the condition of Lorentz invariance of the Lagrangian L translates tothe condition that the terms Z n ν transform as four vectors under Lorentz transformations.Now for a given n Z µn is formed by differentiating the Lagrangian with respect to spacetimederivatives of the field φ n . Therefore Z µn must contain spacetime derivatives of the fielditself. Therefore to transform similarly as W µn , it must be that W µn = kZ µn (11)where k is a scalar which may be a function of the fields and their derivatives. This relationtherefore encodes the Lorentz invariance of the Lagrangian.Now let us proceed to construct the Hamiltonian: H = X n ∂L∂ ( ∂ φ n ) ∂ φ n − L − L = X n (cid:18) ∂L ∂ ( ∂ φ n ) ∂ φ n − ∂L ∂ ( ∂ µ φ n ) ∂ µ φ n (cid:19) − L (12)= − L + X n (cid:18) ∂L ∂ ( ∂ φ n ) ∂ φ n + ∂L ∂ ( ∂ i φ n ) ∂ i φ n (cid:19) (13)= − L + X n δ µν W µn Z νn (14)As η µν W µn Z νn was invariant under Lorentz transformations, it follows that δ µν W µn Z νn wouldbe invariant under orthogonal transformations. Let us convert these into functions of π n φ n . We define F n ≡ W n = ∂ φ n = ∂ H∂π n (15) F in ≡ W in = ∂ i φ n (16) G n ≡ Z n = ∂L ∂ ( ∂ φ n ) = π n (17) G in ≡ Z in = ∂L ∂ ( ∂ i φ n ) = ∂H∂ ( ∂ i φ n ) (18)Where we have used one of the Hamilton’s equations of motion in the first step. Now ifwe decompose the Hamiltonian into H and H using the same logic as we used for theLagrangian, we will have H = L and the Hamiltonian may be written as H = H + X n δ µν F µn G νn (19)The condition (11) for Lorentz invariance of the Lagrangian becomes, in terms of phasespace functions, exactly the condition (5) advertised before: F µn = kG µn (20)Thus for the given class of systems, the relativistic invariance of the dynamics can be readoff from the Hamiltonian by constructing the column matrices F µn and G µn and checking if(20) is satisfied. B. A couple of examples
As a first example let us consider the Hamiltonian of (1). We know from (2) that thisis Lorentz invariant. Now let us check whether it satisfies our criteria. For the A field therelevant vectors are F µn = ( ∂H∂π A , ~ ∇ A ) = (cid:18)(cid:18) gB g B (cid:19) π A , ~ ∇ A (cid:19) (21) G µn = ( π A , ∂H∂ ~ ∇ A ) = (cid:16) π A , (cid:16) g B (cid:17) ~ ∇ A (cid:17) (22)It is easy to check that for this case F µn = kG µn (23)6ith k = (cid:16) g B (cid:17) − Let us consider another example, this time from polymer quantization of scalar fields [3].That this system is not Lorentz invariant is known from its path integral formulation [4].Here we have the following Hamiltonian: H = 12 (cid:0) π + ( ∇ φ ) cos ( µφ ) (cid:1) (24)where µ is a dimensionless quantity. Let us consider the vectors for this case: F µn = ( π, ∇ φ ) (25) G µn = ( π, ∇ φ ) cos ( µφ )) (26)Clearly these are not proportional to each other. Thus our method agrees with the knownresult. C. Encoding Relativistic Invariance in the Hamiltonian: more general case
In this section we briefly sketch how to obtain the criteria for relativistic invariance forsystems satisfying the criteria (i) and (ii) only.The Lagrangian for such a system may be divided into L and L as before and L maybe expanded as: L = 12 ∂L ∂ ( ∂ µ φ ) ∂ µ φ + 14 ∂ L ∂ ( ∂ µ φ ) ∂ ( ∂ ν φ ) ∂ ν φ∂ µ φ + ... (27)This may be written as L = 12 η µν A µ B ν + 14 η αβ η γχ C αγ B β B χ + .... (28)where A µ = ∂L ∂ ( ∂ µ φ ) (29) B µ = ∂ µ φ (30) C µν = ∂ L ∂ ( ∂ µ φ ) ∂ ( ∂ ν φ ) (31)and so on. Again this expression is manifestly Lorentz invariant, given that A µ ∝ B µ , C µν ∝ B µ B ν ... by the same logic as before. Now all these terms may be expressed in the Hamilto-nian formulation in terms of H, π, ~ ∇ φ and derivatives of H to obtain the conditions in theHamiltonian language, just as before. 7 II. SUMMARY AND OUTLOOK
In this paper we have shown that, for a class of systems, there exists a simple wayto check for Lorentz invariance directly from the Hamiltonian formulation. This class ofsystems were defined by conditions (i) and (ii) given above. For an even more restrictedclass, defined by conditions (i), (ii) and (iii) given in section II, the condition reduces toa simple proportionality between two column matrices which may be constructed from theHamiltonian.This is particularly important in the context of quantum theories where other methodsof checking for Lorentz invariance may be immediately available. One place where this isthe case is polymer quantized field theory. Here the only check on Lorentz invariance so farhas come from the path integral formulation [4]. We managed to reach the same conclusionin a much simpler way here.
Acknowledgments
We thank James Edwards for illuminating discussions and helpful comments and GauravNarain for helpful comments on the draft. [1] W. Greiner and J. Reinhardt,
Field Quantization (Springer, New York, 1996).[2] J. Bernstein, Rev. Mod. Phys. , 7 (1974) [Rev. Mod. Phys. , 259 (1975)] [Rev. Mod. Phys. , 855 (1974)]. doi:10.1103/RevModPhys.46.7[3] A. Ashtekar, J. Lewandowski and H. Sahlmann, Class. Quant. Grav. (2003) L11 [gr-qc/0211012].[4] N. Kajuri, Int. J. Mod. Phys. A , 1550204 (2015) doi:10.1142/S0217751X15502048[arXiv:1406.7400 [gr-qc]]., 1550204 (2015) doi:10.1142/S0217751X15502048[arXiv:1406.7400 [gr-qc]].