Quark Gluon Plasma as a Critical State of de Sitter Geometries Leonardo Chiatti a,*, Ignazio Licata b,c a
Medical Physics Laboratory, Health Local Authority,Via Enrico Fermi 15, 01100 Viterbo, Italy
b
Institute for Scientific Methodology, Via Ugo La Malfa 153, 90146 Palermo, Italy School of Advanced International Studies on Applied Theoretical and Non Linear Methodologies of Physics, 70100 Bari, Italy c
Abstract A new type of duality is presented, which connects Quantum Chromo-dynamics (QCD) and de Sitter geometry with the old concept of hadronic “string”. Some applications to hadronization and the origin of Regge trajectories are discussed. The involved global-local connection is explained in term of downward formal causality. Keywords: string theory, de Sitter duality, quark-gluon plasma, R processes, chronon PACS Numbers (2010): 03.65 Ta; 12.40 Nn; 12.38 Mh; 11.25 Tq
1 Introduction Without any doubt, among the challenges of 20th century physics the exploration of hadronic world was the one that more pushed theoretical research towards foundational questions, enforcing a critical rethinking of the concept of "particle" and "interaction". The majority of the problems raised are still being tested in the QCD and Standard Model (SM) [1, 2, 3]. In particular the string theory, derived from dual resonance models, is the most complex and fascinating mathematical result of a journey that today seems to go well beyond strong interactions, towards more ambitious goals of unification [4, 5]. In this paper, we bring string theory back to its original roots, suggesting a unifying perspective of hadron models based on a new type of holographic projection principle. De Sitter's Projective Relativity has been used since the '80s as a method for the construction of classical unified theories [6, 7, 8]. It was recently used to build a group approach to quantum cosmology [9, 10, 11, 13]. In this theory, the big bang is described as a nucleation of R processes from a 5-dimensional de Sitter pre-space, through a Wick rotation1. The structure of this theory sharply constrains the cosmological model and sets the cosmological constant by means of a thermalized complex time vacuum. A structurally related approach has recently been proposed in microphysics [13, 14]. It emphasizes the crucial role of a particular temperature in the localization process of elementary particles, whose value is of the order of the well-known Hagedorn temperature [15]. This work reconsiders various aspects of hadron phenomenology, by analyzing the structural constraints that an approach of this kind poses on the vertices of interaction and spatio-temporal manifestations of a "particle" (Sec. 2). This idea seems to accept the "bootstrap" suggestion of the *
Corresponding author. E_mail addresses:
[email protected] (Chiatti),
[email protected] (Licata). The term “R process” is used here as described by R. Penrose to indicate the localization, and generally the manifestation, of an elementary physical micro-event. The terminology is well known to physicists who deal with the foundations of quantum mechanics, but it is rather uncommon among particle physicists, a field where it should instead have a very natural citizenship, as we shall see. 1
old S-matrix theory [17,18,19,20]; however, the essential difference, as we shall see, lies in specifying a vertex structure in a manner completely consistent with Quantum Field Theory (QFT). More specifically, alongside the "ordinary" QFT vacuum, we postulate a logically precedent pre-spatial structure that binds its dynamic manifestations. This will allow us to revise the extended/point-like dichotomy that runs through the entire history of hadron physics, introducing a de Sitter duality that acts as a new type of T-Duality (Sec. 3). It will allow us to shed light on a new connection between hadronic micro-universe models traditionally based on strong gravity, or on Finsler geometry (see for ex. [20, 21, 22]), and salient features of string models, two model families thus far viewed as mutually incompatible. This new connection is provided by the projection principle, which connects dS geometry with the string QCD, also providing a confinement model (Sec. 4). The connection between the two model classes (Sec. 5-6) emerges from strict geometrical and physical motivations, related to the adoption of the chronon [22, 23]. The introduction of this time horizon, in accordance with experimental data on strong and electroweak interactions, implies a choice in relation to Poincaré's invariance: we can either choose to interpret the chronon as an indicator of oscillatory behaviour (string), or choose a purely geometric description of the hadron's "interior". The projection principle connects the two descriptions, justifying the equivalence of the two approaches and providing a new basis for dS/QCD hypotheses. The deconfined state of quark-gluon plasma (QGP) is governed by quantization rules made possible by the existence of the chronon. This allows for a reconsideration of traditional arguments such as the origin of Regge trajectories and the geometric meaning of hadronization processes (Sec. 7).
2 Bootstrapping QFT: What about "elementarity"? Quantum field theory (QFT) is usually introduced using the second quantization formalism, which is based on the algebra of creation/annihilation operators for fields associated with various "elementary particle" types. However, the notion of elementariness can be understood in at least two distinct ways. A single core interacting with other fields can be defined as "elementary"; in this sense, quarks are elementary entities and hadrons are compound entities. Alternatively, we can define as "elementary" those entities belonging to a given class which are converted, through mutual interactions, in others entities belonging to the same class, regardless of the amount of energy exchanged. In this latter sense, hadrons are elementary and quarks are charged substructures of hadrons. Leptons are elementary in both senses. Particles that are elementary in the former sense (quarks, leptons and gauge quanta) can be termed "super-elementary", whereas "physical" elementary particles belong to the latter (hadrons, leptons and gauge quantums). The following shall focus on hadrons. Hadrons are therefore characterized by a twofold, quite peculiar elementariness, as exemplified by the problem of confinement. It is thus necessary to adopt a viewpoint in which the local superelementary level emerges in a non-dynamic manner (i.e. atemporal) from the global "physical" level. This, along with the usual agent causality associated with dynamics, which is well-known in physics, entails also the consideration of a downward formal causality, in whose context dynamics appear as a phenomenon of radical emergence or bootstrap. For instance, from this point of view a strong interaction process consisting of an exchange of quarks, of the type: (u, anti-d) + (s, anti-u) → (u, anti-u) + (s, anti-d) is interpreted as an instantaneous rearrangement (in accordance with relativistic invariance) of the quarks involved. Since each quark occupies its own spatial position, this means the process is not
local. However, this "recombinatory" non-locality does not imply deviations from the principle of agent causality and not influences the dispersion relations. When a given physical particle is created (annihilated), it enters into (is removed from) the spacetime continuum. At this very moment, its super-elementary fermions are materialized. We can therefore speak of an "inner space" of the physical particle, as a space defined by the relative coordinates of its internal events. At this point, a remarkable observation is that although the creation/annihilation event (the equivalent of what in quantum mechanical terminology is a "state reduction") is instantaneous, the interaction that causes it isn't at all so. In fact, a finite energy is exchanged in this interaction, and thus according to the Heisenberg uncertainty principle, its duration must also be finite. In essence, to say that the created/destroyed particle possesses a rest energy mc2 means that an interaction which localizes the particle with maximum accuracy has a duration that does not exceed θ = ħ/mc2, a value that depends on the particle. We can thus associate with the particle's internal processes (at the time of its spatio-temporal localization with maximum accuracy) a maximum duration of θ. This means that indicating as x, y, z, t the space and time intervals, respectively, between two events inside the particle, the time proper interval constructed from these intervals should not exceed θ. In other words: c2t2 – x2 – y2 – z2 ≤ c2θ2 ,
(1)
which must also hold for all the particle's "internal observers". However, this means the coordinates x, y, z, t can be chosen somehow, provided the value of θ remains unchanged. In other words, only coordinate transformations that leave the (1) unchanged are allowed. It's easy to see how these transformations form a de Sitter group, and how the "inner space" is therefore a de Sitter space with a radius of cθ [see Appendix I for details]. Naturally, in the case of leptons, coordinates can be chosen such that x = y = z = 0 and the separation of events is only temporal. In this sense, we can say that leptons are purely temporal de Sitter "micro-universes", with a de Sitter time of θ. On the other hand, hadrons have genuinely spatio-temporal "micro-universes" with a radius of cθ. Experimentally, the result is cθ ≤ cθ0 ≈ 10-13 cm, the well-known scale of elementary particles, common to both leptons (classical electron radius) and hadrons (effective radius of a strong interaction). We assume that cθ0 is a new fundamental constant of nature. Elsewhere, we have provided arguments supporting the identification of this constant with the classical electron radius [13]. The introduction of this constant allows us to invert the argument, by associating to each physical particle a given number z ≥ 1, which is a function of the flavour composition of the physical particle and its spin state. This physical particle is associated with a de Sitter microuniverse with a radius of cθ0 /z, and a "skeleton mass" of ħz/θ0 . The attribution of this mass (with the sole exceptions of neutrinos and the electron, see [13; 24]) defines the initial core of the particle's inertia; the physical particle's effective mass is the sum of the skeleton mass with contributions from the internal degrees of freedom and self-interaction. Determining the number z from algebraic and topological considerations related to abstract graphs that can be connected to any given physical particle ("glyphs", see [25]) is a problem currently under investigation, and will be the subject of a separate paper2. We will now focus on the physical particle's spatializaton process, occuring in an interaction vertex in which it is localized with maximum accuracy. The introduction of a fundamental constant, or "chronon", θ0 allows for a definition of a spatial scale cθ0, energy scale (ħ/θ0), and moment of inertia ħθ0. And naturally, a mass scale (ħ/c2θ0). It is remarkable that the energy scale ħ/θ0 = 70 MeV is of the same order of magnitude as the Hagedorn temperature, which amounts to about 160 MeV. 2
Glyphs, among other things, enclose information on the explication process of super-elementary fermions for a given physical particle (see Sec. 3).
We geometrically interpret the creation/annihilation of a physical particle as a geodetic (or "gnomonic") projection of an "internal" de Sitter space-time on the external space-time, with local preservation of the light cone. As is well known, this projection produces a Castelnuovo space-time (Beltrami representation) in which the proper time is limited according to (1), while the contemporaneousness space of the fixed point extends to infinity. However, in the original de Sitter space-time this space is closed. Note that, in a closed space, field equations that satisfy the Gauss theorem inexorably lead to the conclusion that all charges of that field and present in that space are null. Thus, moving from a closed space to an open space is equivalent to the creation of charges (or, conversely, to their destruction if the process is viewed from the opposite direction, i.e. as a back projection). The projection therefore implies the creation of charges, which interact with each other and (due to the finite duration of the interaction and thus the appearance of virtual effects by virtue of the uncertainty principle) with themselves as well. This is the interaction associated with the projection, so that self-interaction exists only as a concomitant phenomenon. The strong interaction radius for the "projected" charges (if they are colour charges) is cθ0. We now need to take a look at the strong hadron processes involving the exchange of quarks (and/or the creation/annihilation of quark-antiquark pairs). The hadron micro-universes entering a vertex of interaction have certain radii, which are generally different from those of hadrons emerging from such a vertex. The reorganization of quark configurations therefore requires an intermediate state in which no definite radius exists. This assumption is equivalent in practice to a hypothesis of deconfinement, resulting in a transition to the quark-gluon plasma (QGP) phase. We can postulate that this phase corresponds to a situation in which, in reality, hadrons individual are still present with individual de Sitter radii. However, let's assume that in this condition quark and gluon exchanges are possible between hadrons, and that the gluon radius of interaction is coincident with the radius of action of the recombination phenomena, and equal to cθ0 . The recombination of quarks and genesis of new micro-universes whose radii are the result of such a recombination corresponds to the phenomenon of hadronization. The hadronization can thus be described as a geometry transition. Note that, as we have seen, the radius cθ0 is associated with a temperature of the same order as the Hagedorn temperature. This is a well-known scenario which corresponds to the transition from free hadron gas to QGP, or vice- versa, occuring at that temperature. We have now sufficient elements to propose a dS/QCD correspondence inspired by the intuition of Lobačevskij (see ref. [26], in particular page 62, with footnote by editor and translator Prof. Lombardo Radice), according to which, as a consequence of the structure of fundamental forces, hyperbolic geometry can describe the structure of space at small scale. The new holographic principle sought is implemented using Projective Relativity, connecting the "written" physical information on de Sitter space with hadron phenomenology.
3 De Sitter Duality We'll now propose a mechanism for genesis (intended in a non-dynamic sense as explication, see [27, 28, 29, 30, 31]) of colour charges. Unlike electric or weak charges, which can be manifested as isolated charges, colour charges must necessarily be expressed in groups of total white colour. From our viewpoint, this mechanism must be non-local, and must inter alia constitute the essence of the relationship of belonging for quarks to a given hadron. Consider the well-known expression of kinetic energy for a particle of mass m in Special Relativity:
T
m c2 v 1 c
2
m c2 .
(2)
As is well known, the last term represents the particle rest energy, v being the particle speed relative to an inertial observer. This is therefore the relative speed of a pair of points, one of which is the origin of the frame of reference. We can consider it as the distance in velocity space. The idea we are proposing is that a pair of colour charges originates, in Castelnuovo's microuniverse, through a spatialization process that is energetically described by an expression which is the dual of eq. (2). In this expression, the "distance in velocity space" is replaced by the "distance in positional space," i.e. the distance tout court. The two points, which previously might also have coincided in space3, are thus separated by this distance. The kinetic energy, on the other hand, is no longer dependent on speed, but rather on the position, so becoming a potential energy. We thus have a pair of separate points endowed with a potential energy, instead of a pair of coinciding (if any) points endowed with a kinetic energy. A pair of charges has been generated. The first substitution to apply to eq. (2) is the following: m c2
S ,0
,
0
(3)
where αS, 0 is a constant. Let us consider, in Castelnuovo's plane (x, t), the line of universe of constant speed v passing through the origin. It will intercept the de Sitter horizon of equation t = r/c (where r = cθ0 is the de Sitter radius) at the distance x = vr/c. Through this relation, made possible by the existence of a de Sitter horizon in Castelnuovo's space, one can relate the relative velocity v to the distance x. The following further substitution can thus be applied for eq. (3): v c
x . r
(4)
Through this substitution we pass from the slope of the line v = constant (the space of the relative velocities) to the distance x (the space of the relative positions). Eq. (2) becomes:
U
S ,0
0
x 1 r
S ,0
2
0
.
(5)
At the denominator in the first term the following proper time appears: T0 0
x 1 r
2
.
(6)
It takes on the maximum possible value on the Castelnuovo chronotope, i.e. θ0, for x = 0 and instead it vanishes on the light cone where x = r. We assume that the charges coupling energy is given by (5), reinterpreting x as the spatial distance between these charges (quarks). For the sake of simplicity these considerations omit the colour SU(3) matrices, whose introduction is, in any case, consequential. 3
As a matter of fact, they coincide exactly if eq. (2) is interpreted within the context of Projective (or de Sitter) Relativity, rather than the usual Special Relativity.
The "force" acting between the two quarks is then expressed as:
F
dU dx
c
S ,0
2
x r
x 1 r
2
,
(7)
provided that: x 1 r
c 0
2
cT0 .
(8)
As is evident, this force is attractive. Moreover, F tends to zero as x → 0 (asymptotic freedom), and grows to infinity for x → r (confinement). Equation (7) can be arranged in the Coulomb form: F
c S
,
2
(9)
posing: x r
S S ,0
x 1 r
2
.
(10)
A coupling constant is thus obtained which is dependent on the distance; it vanishes for x → 0 and diverges for x → r. Taking into account the substitution (10), the potential energy (5) becomes: U
S ,0 c
S ,0
0
(11)
which, aside from a non-essential redefinition of zero in the energy scale, is the classic Coulomb expression in the "distance" ξ. Denoting with g the strong charge for each quark, and applying the well-known definition: S ,0
g2 c
,
(12)
what we've described is the genesis of the two charges g in a pair [as remarked, the product of GellMann colour matrices is here omitted]. These charges stem from "mass" (3), and do not appear to be dependent on the quark colours or flavours. In other words, the proposed mechanism generates a coupling which is invariant for both colour and flavour transformations. The SU(N) flavour invariance is however subsequently broken by hadronization, which assigns to each hadron its own specific de Sitter radius, dependent upon the flavour and spin composition. The Coulombian form of eq. (11) allows for the construction of relativistically covariant field equations that admit it as a possible solution. A QCD theory thus becomes possible. However, the ξ
is not the spatial distance between quarks, since the field quanta (gluons) will be absolutely special entities. The pair of charges and their field has been generated by a non-local downward causation process, so it is expected that local descriptions such as gauge theories are such only in appearance. In particular, it is evident that eq. (11) is actually no more associated with a field than is the kinetic energy (2). Since there was no field acting between the first pair of points, there is none in the second. The kinetic energy as well as the potential energy derived from it by means of duality transformation are properties of the relationship of the two respective pairs of points (as is the spatial or chronological distance of two events), and not a physical agent. Therefore, the "gluon exchange" used to describe the quantized version of the coupling between the two points is limited exclusively to them. Just as the kinetic energy of the original pair of points is not something that can be shared with others, but merely a specific property of that pair. In other words, the gluons are exchanged only between the two quarks. The quarks cannot irradiate them in free space, nor absorb gluons from free space. From a naive point of view, it is as if the gluons were constrained to move on a rail, with the two quarks as end points. The rail is in effect the "flux tube" of the old string models, the Susskind elastic ribbon or Nambu rigid bar. For this reason, we will use the term "string" for the pair of quarks within which this relationship is manifested. In this sense, the quarks are the ends of a string. The string is therefore the projection, on the contemporaneousness space of each of the two quarks, of a segment having that quark and a point-event on its de Sitter horizon as its ends. The projection of this point-event is the other quark. The string potential energy is the kinetic energy the other quark would have, in the frame of reference of the first quark, if the projected point-event belonged to the line of universe of the first quark, thanks to the action of an appropriate boost. The "rest mass" is the coupling energy corresponding to a boost of zero speed, and is expressed by the product of an adimensional constant for the unit of energy (ħ/θ0). In essence, what we've described is an algebraic-geometric operation from which one point engenders two points.
4 Projective Confinement Let's now look at eq. (10). This equation provides us with the behaviour of the strong coupling constant as a function of the distance. Its origin is quite obvious. When describing the inseparability of both quarks using the language of a "force" mediated by gluons, proper time T0 can be considered as the time interval necessary for the exchange of the action αS, 0ħ. When the quarks coincide, this interval reaches the maximum value and the coupling is zero; when the point-event whose projection represents one of the quarks is positioned on the light cone of the other quark, this interval vanishes and the coupling becomes infinitely intense. In order to compare eq. (10) with experimental data, we need to find a relation between the variable y = x/r and the interaction energy E ≈ ħc/l exchanged with an external probe. According to the uncertainty principle (and the Fourier transform), this means that we observe details, within the Castelnuovo micro-universe, of spatial extension l, as we define it in our world. We are interested in the high energy limit, where r = cθ0 (fusion of geometries) and x
