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Singular Hamiltonians in Models with Spontaneous Lorentz Symmetry Breaking

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Singular Hamiltonians in Models with Spontaneous Lorentz Symmetry Breaking PHYSICAL REVIEW D 100, 065017 (2019) Singular Hamiltonians in models with spontaneous Lorentz symmetry breaking Michael D. Seifert* Department of Physics, Astronomy, and Geophysics, Connecticut College, 270 Mohegan Avenue, New London, Connecticut 06320, USA (Received 15 May 2019; published 23 September 2019) Many current models which “violate Lorentz symmetry” do so via a vector or tensor field which takes on a vacuum expectation value, thereby spontaneously breaking the underlying Lorentz symmetry of the Lagrangian. One common way to construct such a model is to posit a smooth potential for this field; the natural low-energy solution of such a model would then be excepted to have the tensor field near the minimum of its potential. It is shown in this work that some such models, while appearing well posed at the level of the Lagrangian, have a Hamiltonian which is singular on the vacuum manifold and are therefore ill posed. I illustrate this pathology for an antisymmetric rank-2 tensor field and find sufficient conditions under which this pathology occurs for more general field theories. DOI: 10.1103/PhysRevD.100.065017 I. INTRODUCTION Many (though not all) of the above-cited models share The prospect of finding new physics via Lorentz sym- two features. First, they accomplish the spontaneous break- metry violation has been of significant interest over the past ing of Lorentz symmetry by assigning a potential energy couple of decades. In many such models, Lorentz sym- VðΨÁÁÁÞ to some Lorentz tensor ΨÁÁÁ. This potential is metry is broken spontaneously; one postulates the existence constructed in a “Higgs-like” way, with a set of minima of a new fundamental field that is not a Lorentz scalar and forming a vacuum manifold in field space. Since the field assigns dynamics to this field that obey Lorentz symmetry ΨÁÁÁ is a Lorentz tensor but we want the underlying but lead it to take on a nonzero “vacuum expectation value.” equations of motion to obey Lorentz symmetry, we have The existence of this nonzero Lorentz vector or tensor field to construct the potential out of one or more Lorentz scalars then provides a preferred geometric structure in spacetime. that are dependent on ΨÁÁÁ. The vacuum manifold will be In effect, such a field would provide a “hook” upon determined by a set of conditions on these Lorentz scalars. which one can “hang” frame-dependent effects. In the For example, if we wish to construct a model in which a presence of a Yukawa-like couplings between this new field Lorentz vector field Aa spontaneously breaks Lorentz and conventional matter fields, the results of experiments symmetry, it is not hard to see that the potential VðAaÞ would depend on the relative orientation in spacetime of the must be some function of the 4-vector norm A Aa, and the ’ a observer s 4-velocity and the new field, leading to frame- vacuum manifold will be the set of all 4-vectors of a dependent effects. Such a field is frequently called a particular norm. “ ” Lorentz-violating field, though this is something of a Second, the kinetic terms in the Lagrangians for these misnomer; the postulated field would still transform theories must be constructed with care. Assuming that the between frames via the standard Lorentz transformation underlying Lagrangian is second order and gives rise to laws. The dynamics of such fields in flat spacetime have second-order equations of motion, the kinetic terms will been studied in their own right [1–4] and similar models typically involve a contraction of the spacetime derivative have also been developed in the context of curved space- ∇ Ψ with itself. However, one cannot usually simply – a b1ÁÁÁbn time as possible modifications to general relativity [5 7]. ∇ Ψ ∇aΨb1ÁÁÁbn write down a kinetic term of the form a b1ÁÁÁbn and call it a day. Since the Minkowski metric is indefinite, such a kinetic term will lead to terms with the “wrong sign” *[email protected] of the kinetic energy in the Hamiltonian, which will generically lead to instabilities in the classical solutions. Published by the American Physical Society under the terms of Instead, the kinetic term is usually constructed out of more the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to complicated combinations of the field derivatives, such that the author(s) and the published article’s title, journal citation, the problematic terms in the kinetic energy do not appear and DOI. Funded by SCOAP3. (via cancellations between various contractions). Again, 2470-0010=2019=100(6)=065017(12) 065017-1 Published by the American Physical Society MICHAEL D. SEIFERT PHYS. REV. D 100, 065017 (2019) returning to the case of a 4-vector field for illustration, it is II. HAMILTONIAN FIELD THEORY 1 ∇ A ∇aAb not hard to see that a kinetic term of the form 2 a b In classical mechanics, the construction of a Hamiltonian 2 _ 2 1 _ ⃗ _ will include terms of the form 2 ½A0 − A , which is from a Lagrangian Lðqα; qαÞ is a relatively straightforward indefinite. However, the time derivatives that arise from process. One performs a Legendre transform on the “Maxwell” kinetic term − 1 F Fab (with F ¼ 2∇ A ) Lagrangian, by defining the conjugate momenta pα ¼ 4 ab ab ½a b ∂ ∂ _ are all of the same sign, evading this problem. L= qα; inverting these relationships to write the velocities _ The price one pays for this good behavior, however, is qα in terms of the coordinates qα and the momenta pα; that the time derivatives of some fields do not appear at all and finally writing the Hamiltonian as Hðqα;pαÞ¼ _ − _ _ in the Lagrangian. This leads to the model being con- pαqα Lðqα; qαÞ, now viewing qα as a function of qα strained; certain canonical field momenta must vanish and pα. One can then find the evolution of the coordinates ’ identically, and so one cannot freely specify both the and momenta via Hamilton s equations, or via the Poisson values of the fields and their time derivatives at some bracket, initial moment t0. df The purpose of this work is to illustrate and explore ¼ff; Hgð1Þ a potential incompatibility between these two features. dt Specifically, a model which contains a constrained field where for any quantities f and g we have with a vacuum manifold, while appearing well posed at the level of the Lagrangian, may in fact have a Hamiltonian that X ∂f ∂g ∂g ∂f becomes singular on the vacuum manifold. This raises ff; gg ≡ − : ð2Þ ∂qα ∂pα ∂qα ∂pα serious doubts about the viability of such a theory; it α predicts that the field, if perturbed slightly from its vacuum In field theory for some set of fields ψ α, one would like manifold, could not smoothly evolve back to the vacuum to follow the same procedure, starting from a Lagrange manifold. This conflict does not occur in all Lorentz- density Lðψ α; ψ_ α; ∇ ψ αÞ. However, an important difference violating field models but instead seems to arise when the i “ ” can arise when one attempts to perform the Legendre number of constraints exceeds the codimension of the transform. It can happen that, due to the structure of the vacuum manifold in field space. kinetic terms, the field velocities cannot all be written in The work is structured as follows. A brief summary of terms of the field momenta. In general, this implies that one the techniques of Hamiltonian field theory pertinent to or more of the equations of motion for the field are actually this result is presented in Sec. II. Section III contains an constraint equations, not evolution equations. illustration of how a singular Hamiltonian can arise in a One can, however, still attempt to construct a Lorentz-violating field theory, by examining the dynamics Hamiltonian that generates the time evolution of the system of a rank-2 antisymmetric tensor field Bab in a Lagrangian [in the sense of (1)] via a construction due to Dirac and that is designed to spontaneously break Lorentz symmetry. Bergmann [8]. A brief description of the method can also Finally, Sec. IV discusses how this pathology could arise in be found in a paper by Isenberg and Nester [9], and the a general field theory. notation used here will largely follow the notation in that ℏ Throughout this work, I will use units in which ¼ work. The construction proceeds as follows: 1 − c ¼ ; the metric signature will be ð þþþÞ. Roman (i) Define the field momenta via the natural generali- … indices a; b; c; will be used to denote spacetime tensor zation: indices; i; j; k; … will be used to denote spatial indices, where necessary. Greek indices α; β; γ; … will generally ∂L πα ¼ : ð3Þ only be used to denote indices in field space. All expres- ∂ψ_ α sions involving repeated indices (either tensor indices or field-space indices) can be assumed to obey the Einstein One can then attempt to invert these relationships to summation convention, unless explicitly stated otherwise; find ψ_ α as a function of πα, ψ α, and their derivatives. in particular, for a significant fraction of Sec. IV B and in However, it may occur that certain equations (or the Appendix, the field-space summations will be written combinations of equations) in (3) do not contain any out explicitly. While I will be working exclusively in the velocities. These equations must be thought of as realm of flat spacetime, I will still use the symbol ∇a constraining the initial data and can be written in (or ∇i) to denote spacetime (or spatial) derivatives; the the form symbol ∂ will be reserved for partial derivatives of ⃗ functions of fields (such as the field potential energy or ΦIðψ α; ∇ψ α; παÞ¼0 ð4Þ the Lagrangian density) with respect to their arguments.
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