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Unparticles” and Exotic “Unmatter” States Volume 4 PROGRESS IN PHYSICS October, 2008 On Emergent Physics, “Unparticles” and Exotic “Unmatter” States Ervin Goldfain and Florentin Smarandachey Photonics CoE, Welch Allyn Inc., Skaneateles Falls, NY 13153, USA E-mail: [email protected] yChair of Math & Sc. Depart., University of New Mexico, Gallup, NM 87301, USA E-mail: [email protected] Emergent physics refers to the formation and evolution of collective patterns in systems that are nonlinear and out-of-equilibrium. This type of large-scale behavior often de- velops as a result of simple interactions at the component level and involves a dynamic interplay between order and randomness. On account of its universality, there are credi- ble hints that emergence may play a leading role in the Tera-ElectronVolt(TeV) sector of particle physics. Following this path, we examine the possibility of hypothetical high- energy states that have fractional number of quanta per state and consist of arbitrary mixtures of particles and antiparticles. These states are similar to “un-particles”, mass- less fields of non-integral scaling dimensions that were recently conjectured to emerge in the TeV sector of particle physics. They are also linked to “unmatter”, exotic clusters of matter and antimatter introduced few years ago in the context of Neutrosophy. 1 Introduction long as " 1. Full scale invariance and equilibrium field theory are asymptotically recovered in the limit of physical space-time (d = 4) as " ! 0 or ! 1 [11, 12]. Quantum Field Theory (QFT) is a framework whose meth- ods and ideas have found numerous applications in various domains, from particle physics and condensed matter to cos- 2 Definitions mology, statistical physics and critical phenomena [1, 2]. As We use below the Riemann-Liouville definition for the one- successful synthesis of Quantum Mechanics and Special Rel- dimensional left and right fractal operators [13]. Consider for ativity, QFT represents a collection of equilibrium field theo- simplicity a space-independent scalar field '(t). Taking the ries and forms the foundation for the Standard Model (SM), time coordinate to be the representative variable, one writes a body of knowledge that describes the behavior of all known particles and their interactions, except gravity. Many broken Zt 1 d symmetries in QFT, such as violation of parity and CP in- 0DL '(t) = (t ) '()d ; (1) variance, are linked to either the electroweak interaction or (1 ) dt 0 the physics beyond SM [3–5]. This observation suggests that unitary evolution postulated by QFT no longer holds near or Z0 1 d above the energy scale of electroweak interaction 0DR '(t) = ( ) ( t) '()d : (2) ( 300GeV) [6,7]. It also suggests that progress on the the- (1 ) dt t oretical front requires a framework that can properly handle 0 < < 1 non-unitary evolution of phenomena beyond SM. We believe Here, fractional dimension denotes the order that fractional dynamics naturally fits this description. It op- of fractional differentiation. In general, it can be shown that erates with derivatives of non-integer order called fractal op- is linearly dependent on the dimensionality of the space-time erators and is suitable for analyzing many complex processes support [8]. By definition, assumes a continuous spectrum with long-range interactions [6–9]. Building on the current of values on fractal supports [11]. understanding of fractal operators, we take the dimensional parameter of the regularization program " = 4d to represent 3 Fractional dynamics and ‘unparticle’ physics the order of fractional differentiation in physical space-time (alternatively, " = 1 d in one-dimensional space) [10, 11]. The classical Lagrangian for the free scalar field theory in It can be shown that " is related to the reciprocal of the cutoff 3+1 dimensions reads [1–2, 14] 0 scale " ( /), where 0 stands for a finite and arbitrary 2 2 L = @ '@' m ' ; (3) reference mass and is the cutoff energy scale. Under these " circumstances, may be thought as an infinitesimal param- and yields the following expression for the field momentum eter that can be continuously tuned and drives the departure @L @' from equilibrium. The approach to scale invariance demands = = : @' @t (4) that the choice of this parameter is completely arbitrary, as @( @t ) 10 Ervin Goldfain and Florentin Smarandache. On Emergent Physics, “Unparticles” and Exotic “Unmatter” October, 2008 PROGRESS IN PHYSICS Volume 4 It is known that the standard technique of canonical quan- 4 Mixing properties of fractal operators tization promotes a classical field theory to a quantum field theory by converting the field and momentum variables into Left and right fractal operators (L/R) are natural analogues of operators. To gain full physical insight with minimal com- chiral components associated with the structure of quantum plications in formalism, we work below in 0+1 dimensions. fields [8, 9]. The goal of this section is to show that there is an Ignoring the left/right labels for the time being, we define the inherent mixing of (L/R) operators induced by the fractional field and momentum operators as dynamics, as described below. An equivalent representation of (1) is given by ' ! 'b = '; (5) Z0 1 d 0DL '(t) = ( ) [( t)] '() d ; (14) @ (1 ) dt ! b = i iD : (6) @ j'j t or Without the loss of generality, we set m = 1 in (3). The Z0 (1) d Hamiltonian becomes D '(t) = ( t) '() d = 0 L (1 ) dt 1 1 1 t b 2 2 2 2 H ! H = D + ' = (b + ' ) : (7) 2 2 2 = (1) 0DR '(t) ; (15) By analogy with the standard treatment of harmonic oscil- lator in quantum mechanics, it is convenient to work with the 0DR = (1) 0DL = exp(i ) 0DL : (16) destruction and creation operators defined through [1–2, 14] Starting from (2) instead, we find : 1 ba = p ['b + ib ] ; (8) D = (1) D = exp(i ) D : 2 0 L 0 R 0 R (17) : 1 Consider now the one-dimensional case d = 1, take ba+ = p ['b i b ] : (9) = " = 1 d " 2 and recall that continuous tuning of does not impact the physics as a consequence of scale invariance. Let Straightforward algebra shows that these operators satisfy us iterate (16) and (17) a finite number of times (n > 1) under the following commutation rules the assumption that n" 1. It follows that the fractal opera- tor of any infinitesimal order may be only defined up to an ar- [ba; ba] = [ba+ ; ba+ ] = 0 ; (10) bitrary dimensional factor exp(in") 1+(in") = 1i"e, + ( 1) that is, [ba ; ba ] = i [';b b ] = b : (11) " 0 0DL;R '(t) 0DL;R i"e '(t) (18) The second relation of these leads to or " 0 1 i0DL;R '(t) = i 0DL;R + "e '(t) ; (19) Hb = ba+ ba + b( 1): (12) 2 where " lim DL;R '(t) = '(t) : (20) In the limit = 1 we recover the quantum mechanics of "!0 the harmonic oscillator, namely Relations (18–20) indicate that fractional dimension "ein- 1 duces: (a) a new type of mixing between chiral components Hb = ba+ba + : 2 (13) of the field and (b) an ambiguity in the very definition of the field, fundamentally different from measurement uncertain- It was shown in [6] that the fractional Hamiltonian (12) ties associated with Heisenberg principle. Both effects are leads to a continuous spectrum of states having non-integer irreversible (since fractional dynamics describes irreversible numbers of quanta per state. These unusual flavors of par- processes) and of topological nature (being based on the con- ticles and antiparticles emerging as fractional objects were cept of continuous dimension). They do not have a counter- named “complexons”. Similar conclusions have recently sur- part in conventional QFT. faced in a number of papers where the possibility of a scale- invariant “hidden” sector of particle physics extending be- 5 Emergence of “unmatter” states yond SM has been investigated. A direct consequence of this setting is a continuous spectrum of massless fields having Using the operator language of QFT and taking into account non-integral scaling dimensions called “un-particles”. The (6), (18) can be presented as reader is directed to [15–21] for an in-depth discussion of “un-particle” physics. b"'(t) = b"'(t) "e'b(t) : (21) Ervin Goldfain and Florentin Smarandache. On Emergent Physics, “Unparticles” and Exotic “Unmatter” 11 Volume 4 PROGRESS IN PHYSICS October, 2008 Relation (21) shows that the fractional momentum op- positive particle (say clusters of quarks and antiquarks whose erator b" and the field operator 'b(t) = '(t) are no longer total charge is positive, etc.). independent entities but linearly coupled through fractional dimension "e. From (11) it follows that the destruction and 8 Unmatter atom creation operators are also coupled to each other. As a re- sult, particles and antiparticles can no longer exist as linearly It is possible to define the unmatter in a more general way, "e independent objects. Because is continuous, they emerge using the exotic atom. as an infinite spectrum of mixed states. This surprising find- The classical unmatter atoms were formed by particles ing is counterintuitive as it does not have an equivalent in like (a) electrons, protons, and antineutrons, or (b) antielec- conventional QFT. Moreover, arbitrary mixtures of particles trons, antiprotons, and neutrons. and antiparticles may be regarded as a manifestation of “un- In a more general definition, an unmatter atom is a system matter”, a concept launched in the context of Neutrosophic of particles as above, or such that one or more particles are Logic [22–24]. replaces by other particles of the same charge. Other categories would be (c) a matter atom with where 6 Definition of unmatter one or more (but not all) of the electrons and/or protons are replaced by antimatter particles of the same corresponding In short, unmatter is formed by matter and antimatter that charges, and (d) an antimatter atom such that one or more (but bind together [23, 24].
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