Lifshitz Phase Gauge Symmetry Breaking and Application to $E_6$ and $E_8$ Grand Unification
This paper has been withdrawn, because the argument for generating Goldstone modes at a Lifshitz point is incorrect.
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arXiv subjects
Stephen L. Adler
Publications and source records attributed to Stephen L. Adler.
Completing the Square to Find the Supersymmetric Matter Effective Action Induced by Coupling to Linearized N=1 Supergravity
We consider generic N=1 supersymmetric matter coupled to linearized N=1 supergravity through the multiplet of currents. By completing the square, we find the effective action giving the leading supergravity induced correction to the matter dynamics, expressed explicitly as a quadratic form in the components of the current multiplet. The effective action is supersymmetry invariant through an interplay of the local terms arising from the auxiliary field couplings, and the nonlocal terms arising from graviton and gravitino exchange, neither of which is separately invariant. Having an explicit form for the supergravity induced effective action is a first step in studying whether supergravity corrections can lead to dynamical supersymmetry breaking in supersymmetric matter dynamics. In Appendices we give explicit expressions for the currents, in our notational conventions, in the Wess-Zumino and supersymmetric Yang Mills models.
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Probability in Orthodox Quantum Mechanics: Probability as a Postulate Versus Probability as an Emergent Phenomenon
The role of probability in quantum mechanics is reviewed, with a discussion of the ``orthodox'' versus the statistical interpretive frameworks, and of a number of related issues. After a brief summary of sources of unease with quantum mechanics, a survey is given of attempts either to give a new interpretive framework assuming quantum mechanics is exact, or to modify quantum mechanics assuming it is a very accurate approximation to a more fundamental theory. This survey focuses particularly on the issues of whether probabilities in quantum mechanics are postulated or emergent.
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Scalar Exchange Forces and Generalized Most Attractive Channel Rule
We discuss the possibility that fermionic condensates arise from the dominance of scalar exchange forces over vector gluon exchange. When a scalar in the adjoint representation is exchanged in the reaction $A+A \to A+A$, the usual most attractive channel (MAC) rule is reversed in sign, with the consequence that the formation of a condensate with the largest possible Casimir is favored. More generally, when a scalar in a general representation is exchanged in the reaction $A+B \to B+A$, the group theoretic sum giving the force sign and strength can be expressed in terms of a Racah coefficient for the group in question. We illustrate the formalism in the case of the group SU(2), and give possible applications to SO(10) and $E_6$ grand unification.
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The Equilibrium Distribution of Gas Molecules Adsorbed on an Active Surface
We evaluate the exact equilibrium distribution of gas molecules adsorbed on an active surface with an infinite number of attachment sites. Our result is a Poisson distribution having mean $X = {μP P_s \over P_e}$, with $μ$ the mean gas density, $ P_s$ the sticking probability, $P_e$ the evaporation probability in a time interval $τ$, and $P$ Smoluchowski's exit probability in time interval $τ$ for the surface in question. We then solve for the case of a finite number of attachment sites using the mean field approximation, recovering in this case the Langmuir isotherm.
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Derivation of the Lindblad Generator Structure by use of the Itô Stochastic Calculus
We use the Itô stochastic calculus to give a simple derivation of the Lindblad form for the generator of a completely positive density matrix evolution, by specialization from the corresponding global form for a completely positive map. As a by-product, we obtain a generalized generator for a completely positive stochastic density matrix evolution.
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Structure and Properties of Hughston's Stochastic Extension of the Schrödinger Equation
Hughston has recently proposed a stochastic extension of the Schrödinger equation, expressed as a stochastic differential equation on projective Hilbert space. We derive new projective Hilbert space identities, which we use to give a general proof that Hughston's equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We discuss the relation of Hughston's equation to earlier work on norm-preserving stochastic equations, and show that Hughston's equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix. We discuss the behavior of systems constructed as direct products of independent subsystems, and briefly address the question of whether an energy-based approach, such as Hughston's, suffices to give an objective interpretation of the measurement process in quantum mechanics.
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State Vector Collapse Probabilities and Separability of Independent Systems in Hughston's Stochastic Extension of the Schrödinger Equation
We give a general proof that Hughston's stochastic extension of the Schrödinger equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We also show that for a system composed of independent subsystems, Hughston's equation separates into similar independent equations for the each of the subsystems, correlated only through the common Wiener process that drives the state reduction.
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Higgs Mass Bounds in the Three- and Six-Higgs Doublet Models for Family Structure
We reanalyze our recently proposed mass matrix model based on spontaneously broken discrete chiral family symmetry, taking into account the additional flavor changing neutral current constraint implied by the bound on the $D_1-D_2$ mass difference, and including several corrections to our earlier analysis. When combined, the $K_1-K_2$ and $D_1-D_2$ constraints force the masses of the Higgs particles that contribute most strongly to flavor changing neutral currents (the $ϕ$ Higgs states) to lie above 17 TeV, well beyond the limit of validity of conventional perturbative Higgs physics. The analogous constraints on the masses of the $η$ Higgs states and the neutral pseudo Goldstone Higgs state depend on the mechanism for realizing small first family masses. If the $η$ Higgs is the primary contributor to second family masses, the pseudo Goldstone and $η$ Higgs states must have masses above 220 GeV, with numerical fits suggesting masses above 1 TeV, while if the $η$ Higgs is responsible solely for first family masses, the corresponding mass bounds drop to the range detectable at the LHC. We show that naturalness of small first family masses favors the latter alternative, and give an illustrative mass matrix texture model.
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General Theory of Image Normalization
We give a systematic, abstract formulation of the image normalization method as applied to a general group of image transformations, and then illustrate the abstract analysis by applying it to the hierarchy of viewing transformations of a planar object.
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Model for Particle Masses, Flavor Mixing, and CP Violation Based on Spontaneously Broken Discrete Chiral Symmetry as the Origin of Families
We construct extensions of the standard model based on the hypothesis that the Higgs bosons also exhibit a family structure, and that the flavor weak eigenstates in the three families are distinguished by a discrete $Z_6$ chiral symmetry that is spontaneously broken by the Higgs sector. We study in detail at the tree level models with three Higgs doublets, and with six Higgs doublets comprising two weakly coupled sets of three. In a leading approximation of $S_3$ cyclic permutation symmetry the three Higgs model gives a ``democratic'' mass matrix of rank one, while the six Higgs model gives either a rank one mass matrix, or in the case when it spontaneously violates CP, a rank two mass matrix corresponding to nonzero second family masses. In both models, the CKM matrix is exactly unity in leading approximation. Allowing small explicit violations of cyclic permutation symmetry generates small first family masses in the six Higgs model, and first and second family masses in the three Higgs model, and gives a non-trivial CKM matrix in which the mixings of the first and second family quarks are naturally larger than mixings involving the third family. Complete numerical fits are given for both models, flavor changing neutral current constraints are discussed in detail, and the issues of unification of couplings and neutrino masses are addressed. On a technical level, our analysis uses the theory of circulant and retrocirculant matrices, the relevant parts of which are reviewed.
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Gauge Fixing in the Partition Function for Generalized Quantum Dynamics
We discuss the problem of gauge fixing for the partition function in generalized quantum (or trace) dynamics, deriving analogs of the De Witt-Faddeev-Popov procedure and of the BRST invariance familiar in the functional integral context.
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Corrections to the Emergent Canonical Commutation Relations Arising in the Statistical Mechanics of Matrix Models
We study the leading corrections to the emergent canonical commutation relations arising in the statistical mechanics of matrix models, by deriving several related Ward identities, and give conditions for these corrections to be small. We show that emergent canonical commutators are possible only in matrix models in complex Hilbert space for which the numbers of fermionic and bosonic fundamental degrees of freedom are equal, suggesting that supersymmetry will play a crucial role. Our results simplify, and sharpen, those obtained earlier by Adler and Millard.
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A Strategy for a Vanishing Cosmological Constant in the Presence of Scale Invariance Breaking
Recent work has shown that complex quantum field theory emerges as a statistical mechanical approximation to an underlying noncommutative operator dynamics based on a total trace action. In this dynamics, scale invariance of the trace action becomes the statement $0=Re Tr T_μ^μ$, with $T_{μν}$ the operator stress energy tensor, and with $Tr$ the trace over the underlying Hilbert space. We show that this condition implies the vanishing of the cosmological constant and vacuum energy in the emergent quantum field theory. However, since the scale invariance condition does not require the operator $T_μ^μ$ to vanish, the spontaneous breakdown of scale invariance is still permitted.
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Frustrated SU(4) as the Preonic Precursor of the Standard Model
We give a model for composite quarks and leptons based on the semisimple gauge group SU(4), with the preons in the 10 representation; this choice of gauge gluon and preon multiplets is motivated by the possibility of embedding them in an N=6 supergravity multiplet. Hypercolor singlets are forbidden in the fermionic sector of this theory; we propose that SU(4) symmetry spontaneously breaks to $SU(3) \times U(1)$, with the binding of triality nonzero preons and gluons into composites, and with the formation of a color singlet condensate that breaks the initial $Z_{12}$ vacuum symmetry to $Z_{6}$. The spin 1/2 fermionic composites have the triality structure of a quark lepton family, and the initial $Z_{12}$ symmetry implies that there are six massless families, which mix to give three distinct families, two massless with massive partners and one with both states massive, at the scale of the condensate. The spin 1 triality zero composites of the color triplet SU(4) gluons, when coupled to the condensate and with the color singlet representation of the 10 acting as a doorway state, lead to weak interactions of the fermionic composites through an exact SU(2) gauge algebra. The initial $Z_{12}$ symmetry implies that this SU(2) gauge algebra structure is doubled, which in turn requires that the corresponding independent gauge bosons must couple to chiral components of the composite fermions. Since the U(1) couples to the 10 representation as $B-L$, an effective $SU(2)_L \times SU(2)_R \times U(1)_{B-L}$ electroweak theory arises at the condensate scale, with all composites having the correct electric charge structure. A renormalization group analysis shows that the conversion by binding of one 10 of SU(4) to 12 triplets of SU(3) can give a very large, calculable hierarchy ratio between
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A Rejoinder on Quaternionic Projective Representations
In a series of papers published in this Journal (J. Math. Phys.), a discussion was started on the significance of a new definition of projective representations in quaternionic Hilbert spaces. The present paper gives what we believe is a resolution of the semantic differences that had apparently tended to obscure the issues.
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Poincaré Supersymmetry Representations Over Trace Class Noncommutative Graded Operator Algebras
We show that rigid supersymmetry theories in four dimensions can be extended to give supersymmetric trace (or generalized quantum) dynamics theories, in which the supersymmetry algebra is represented by the generalized Poisson bracket of trace supercharges, constructed from fields that form a trace class noncommutative graded operator algebra. In particular, supersymmetry theories can be turned into supersymmetric matrix models this way. We demonstrate our results by detailed component field calculations for the Wess-Zumino and the supersymmetric Yang-Mills models (the latter with axial gauge fixing), and then show that they are also implied by a simple and general superspace argument.
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The Matrix Model for M Theory as an Exemplar of Trace (or Generalized Quantum) Dynamics
We show that the recently proposed matrix model for M theory obeys the cyclic trace assumptions underlying generalized quantum or trace dynamics. This permits a verification of supersymmetry as an operator calculation, and a calculation of the supercharge density algebra by using the generalized Poisson bracket, in a basis-independent manner that makes no reference to individual matrix elements. Implications for quantization of the model are discussed. Our results are a special case of a general result presented elsewhere, that all rigid supersymmetry theories can be extended to give supersymmetric trace dynamics theories, in which the supersymmetry algebra is represented by the generalized Poisson bracket of trace supercharges, constructed from fields that form a noncommutative trace class graded operator algebra.
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