to model "matter components" (Veneziano)
The string "theory" concept is about replacing the 0-dimensional “particle” "entity/ object" (which is not the same as today’s quantum object model (!)) by a 1-dimensional "string" "entity/ object". It aims to model different kinds of quantum vibrations, which then "generate" "energy", respectively different kinds of "quantum energy states"...
... unfortunately, the simple fact is, that compared to today's particle-field model dualism/paradox this approach generates no additional value for any experimental setup or interpretation of experimental data, nor any value added to existing conceptual challenges, e.g. concerning the "particle-field paradox" inherent “contact point / body” interaction issue.
Every conceptual challenge with respect to synchronize/ integrate GRT and quantum field theory remains unsolved/ not answered, respectively generates even more challenges, as now e.g. the"contact body" problem needs to be solved for a one-dimensional object, than "only" for a "zero" dimensional "object" (...:) ) . Different kinds of (open and closed) "strings" add additonal complexity (finite (!) length, still requiring "end points /particles" for all kinds of those "manifolds", e.g. open intervals, as well as for closed intervals).
The standard model of a quantum x in current quantum mechanics is an element x of a Hilbert space. If one would built a feasible analogue model of a “string-quantum" in current quamtum mechanics framework (a Kind of all-inclusive package for the quantum itself, its location and its vibration/ energy) basically one needs to define something simliar to the already existing wave packages. In current quantum mechanics this leads to the Heissenberg uncertainty relation (respectively "uneigentliche" eigenvectors) , which is just annother view on the particle-field dualism.
In “Braun K., An alternative quantization of H=xp” an alternative distributional Hilbert space framework in combination with Pseudo-Differential Operators are proposed to overcome the above challenge.
Background Information: String "Theory" and "Loop Quantum Gravity"
The superstring theory is based on the idea to replace the “particle” (no extension, no interaction between particles) by a string with extension into one dimension. It enables vibrations and different kind of vibration states.
S. Lie developed the concept of (continuous) contact transformations to provide a mathematical framework (gauge theories) to "solve" the contact body problem for "manifolds" (as model of "multiple extended quantities"). Basically it adds to each manifold (of each multiple extended quantities model per nature force) an appropriate group to enabled "geometric model" properties. Those groups must be "orthogonal" to each others, Therefore the required space dimension increases per added "force model" (and no interaction between those forces are possible):
The Standard (field) Model of Elementary Particles (SMEP) is given by SU(3) x SU(2) x U(1). Its components are the following interaction dynamics fields:
1. Electromagnetic Interaction Dynamics (EID): U(1)
2. Weak Interaction Dynamics (WID): SU(2) x U(1)
3. Strong Interaction Dynamics (SID): SU(3).
A symmetry group to link the superstring field concept with SMEP is still missing:
4. “String Interaction Dynamics” (SUSY: Super Symmetry?)?
à A SUSY field model requires an inflation of space dimension n>10 with no validation against physical standards like the Huygens principle, but also with respect to results concerning the characterization of the space-time dimension n=4. There is no mathematical model existing.
In the context of overcoming the long list of constraints, which jeopardize any solution to build a mathematical model, we quote from
Kaku M., "Introduction to Superstrings and N-Theory”, Springer Verlag, NewYork, 1999
“Because general relativity and quantummechanics can be derived from a small set of postulates, one or more of these postulates must be wrong. The key must be to drop one or more of these assumptions about Nature on which we have constructed general relativity and quantum mechanics. Over the years several proposals have been made to drop some of our common sense notions about the universe: continuity, causality,unitarity, locality, point particles.”
We note that the “Loop Quantum Gravity (LQG)”, as alternative to the super string theory, is also built on a Hilbertspace K(diff), modeling 3D diffeomorphism invariance and transformation properties of spin network states under diffeomorphism ((RoC) 6.4). TheHamiltonian for the fields in a standard analysis framework is defined by ((RoC) 6.4.2)
H := H(Einstein)+H(Yang-Mills) + H(Dirac) + H(Higgs) ,((RoC) 7.3.
(RoC) 1.2.1: “The LQG is characterized by the choice of a different algebra of basisfield functions, as in Quantum Field Theory (QFT). In conventional QFT this isgenerally the canonical algebra formed by the positive and negative frequencycomponents of the filed modes. The quantization of this algebra leads to thecreation and annihilation operators a and a(+). The characterization of thepositive and negative frequencies requires a background space-time. In contrast to this, what characterizes LQG is the choice of a different algebra of basisfield functions: a non-canonical algebra based on the holonomies of the gravitational connection. The holonomy (or “Wilson loop”) is the matrix of the parallel transport along a closed curve.”
This means, that also the LQG is built on the same handicaps, as H. Weyl’s affine geometry. For an alternative approach for a quantum gravity model (with respect to the concepts of“continuity” and “particles”) we refer to
(RoC) Rovelli C., „Quantum Gravity“, Cambridge University Press, 2004
History of string theory
The string theory and the early days of dual models is going back to the "Regge trajectory" and Veneziano amplitude and duality. From (GrM) below we recall:
1.2.1 Duality and the Graviton
One of the difficulties of dual theories of strong interactions, apart from the failure to accommodate the parton properties, was that these models always predicted a variety of massless particles, none of which are present in the hadronic world. These massless particles show up, for instance, in the poles that appear at s=0 and t=0 in the Veneziano amplitude, once one set a(0)=1 in order to eliminate ghosts. Dual models turn out to have massless particles of various spins. In particular, the closed-string sector of dual models turns out to have massless spin two particle. Investigation reveals - as one might suspect on general grounds - that its couplings are similar to those of general relativity. Might we interpret this particle as the graviton?
From a mathematical conceptual point of view it "just" "adds (while keeping orthogonality relationships with the prize to be paid, ending up with a high number of dimensions)" the 4 current arte-facts from gravitation and QED, QCE, QFE field theory, which are
- gluon (QCD) --> color charge
- photon (QED) --> electrical charge
- "Higgs-(...on)" (QFD) --> flavor charge, weak iso spin
- "graviton"--> mass.
The definition of the "Hamiltonian purpose" might be interpreted as "sense", in the way, as it is defined by M. Heidegger (§32, Verstehen und Auslegen, p. 151) as "das, worin sich die Verständlichkeit von etwas hält" ("that, wherein the understandability of something keeps").
M. Heidegger, "Sein und Zeit", Max Niemeyer Verlag Tübingen, 2001.
D. E. Neuenschwander, "Emmy Noether's Wonderful Theorem", 3.1, 3.4
Newton´s law (equations of motion) <---> action principle
Quote: "only when the kinetic (K) and potential (U) energies and the calculus of variations came together could it be realized how F=m*a can be subsumed into the latter, through the Lagrangian K-U."
Kaku M., (KaM):
"This equivalence breaks down at the quantum level. The equations of motion is only an approximation to the actual quantum behavior of matter. Thus the action principle is the only acceptable framework for quantum mechanics."
The Lagrange formalism (building on the physical concepts of "force" and "work") and the Hamiltonian formalism (building on "energy" and "action") are equivalent. The transformation is given by the Legendre transformation. To the author's opinion a GUT theory needs to be built into a mathematical framework, where both formalism are no longer equivalent, i.e. the Legendre transformation is not possible resp. the regularity requirements are not fulfilled. This is, where J. Plemelj could come into the game (currently the "force" is calculated/represented as gradient of a corresponding potential in the usual sense). In this new framework, "force" and "work" are not existing in the sense of Newton (but in a less regular sense, following Plemelj's alternative definition of a potential), but the "energy" and "action" definitions keep valid (energy norm and corresponding minimization problem). If "force" is not required to be defined in the standard way, there is no need to prove the existence of corresponding "test" particle, to which the "force" is acting to (the massless bosons fill anyway 99% of the "volume" of a nucleon in the "presence" of force). From a mathematical point of view the force gradient is currently seen and modeled as a continuous field, while the related energy is accepted and modeled by discrete quantum leaps. This is at least "remarkable" and indicates also a switch to a Stieltjes integral representation of the underlying force potential.
In a nutshell: to the author´s opinion the current conceptual problem to unify the QED, QCD, QFD and gravitation theory is a mathematical problem (defining the adequate framework/Hilbert space environment) and not a problem of physics. It´s already proven, that there are neither particles nor fields in the way the current mathematical framework requires them to describe the existing physical models. The reflections of Hermann Weyl (see below (WeH), introduction) concerning space, time, matter and its "strong", "weak" or "anything else" "interactions" seems to be still valid. H.Weyl ((BlD), (WeH1)) introduced the concepts of gauge transformation and gauge invariance to compute the field strength (or curvature) of gauge potential (or connection). In case G=U(1) (or equivalent, the group of rotations in the plane), the gauge potential is essentially the 4-vector potential of electromagnetism and the field strength is the electromagnetic field (BlD).
note: the group SU(2) (or equivalently, the unit quaternions) corresponds to the gauges prescribing a point-dependent choice of isotopic spin axis (introduced by Yang and Mills).
In order to show that the Veneziano amplitude obeys duality the analytical behavior of the Euler beta function is applied ((GrM), 1.1.1). The beta function especially describes very well the high-energy behavior of the Veneziano model. A string is a given state of oscillation that has a mass m ((GrM), 2.1). The mass squared can be determined in terms of internal modes of oscillation of the string. Both representations, for open and closed strings, correspond to a Laurent series with vanishing "zero" Fourier term. The Gamma and the Beta function play a key role in the Riemann Hypothesis. The approach given in
is basically about the following: Transformation of the constant, non-vanishing Fourier term of the Theta function into a vanishing Fourier term (by applying the Hilbert Transformation to the Theta function). The prize to be paid is change of the framework from analytical function to L(2)-Hilbert space environment. A corresponding approach might lead to an appropriate Hilbert space definition (H(-1/2)?), which, as well, might fit to the proposed Hilbert space environment for the Stefan problem.
We note that the Dirac delta function is an element of H(-s/2) for s>n/2, whereby n denotes the dimension of the field.
Following the road to surface minimization formulation would create a loop back to the Poincare conjecture;
see e.g. Michael T. Anderson, "Geometrization of 3-manifolds via the Ricci flow":
The QCD (Quantum Cromodynamics) is the gauge field theory that describes the strong interactions of colored partons (quarks and gluons). It is the SU(3) component of the SU(3)xSU(2)xU(1) standard model of particle physics. The mathematical formalism is given by the Lagrangian of QCD. The parton distributions are crucial ingredients in the theoretical predictions of scattering cross-sections at hadron colliders. J. Plemelj's concept enables the definition of a parton induced potential, based on partons' (infinitely small) mass, not based on a parton distribution!
The QED field theory describes the weak interactions (the term "force" is no longer needed!) of electromagnetic fields.
A mathematical concept to handle non-linear differential or integral equations is given by K. O. Friedrich's dual (complementary) extremal problem formalism. For example in electrostatic field theory the dual principle to the "principle of minimal potential energy" is the "principle of Thomsen". The "method of Noble" leads to a saddle point problem, described by a system of adjoint operator equations S, T, where S, T are represented as Gateaux derivatives with respect to u resp. v of a Hamiltonian (energy density) function W(u,v).
E.Schroedinger, "Statistical Thermodynamics", Cambridge University Press, 1952
R. P. Feynman, A. R. Hibbs, Quantum mechanics and path integrals, Dover Publications, Inc., Mineola, New York, 2010, pp. 240, 244, 268, 294.
L. B. Rall, On complementary variational principles, J. Math. Anal. Appl. 14 (1966) 174-184,
A. M. Arthurs, Complementary variational principles, Oxford, 1970, chapter 4.
The general concept to define a valid physical model is to define mathematical field model (by (physical) Lagrange principle with corresponding mathematical variational principles), which requires per definition the existence of a (mathematical arte-fact) test particle ("Aufpunkt"). From this physical model e.g. corresponding energy (quantum) values are calculated, which then are compared to experimental measurement results. If the measured values fits to the theoretically calculated "forecast" the (physical!!) existence of the required test particle is assumed to be proven.
Applying the dual pair principles of "energy principles" and its "dual principles" for e.g. Fermions and Bosons in combination with the opportunities provided by "J. Plemelj" "to model potentials just built on infinitely small particles might open alternative way to define quantum fields.
Following our proposal from
for an alternative zero point energy model the appropriate Hilbert space framework modeling the dual (variational) equations should not be H(0), but H(-a), a>0.
We emphasize that our model, representing a quantum as an element of e.g. H(-1/2) Hilbert space, models such a spontaneous breakdown just by applying the projection operator from H(-1/2) into H(0) Hilbert space, resulting into the fact, that a quantum dual operator (which defines the corresponding H(-1/2)-energy (-norm)), "breaks down spontaneously" to an asymmetric operator within the smaller (today´s standard) H(0) Hilbert space.
From the famous paper of P. Higgs we recall in this context:
. ..."the idea that the apparently approximate nature of the internal symmetries of elementary-particle physics is the result of asymmetries in the stable solution of exactly symmetric dynamical equations .... is an attractive one. .... Within the framework of quantum field theory such a "spontaneous" breakdown of symmetry occurs if a Lagrangian, fully invariant under the internal symmetry group, has a structure that the physical vacuum is a member of a set of (physically equivalent) states which transform according the a nontrivial representation of the group. .... That vacuum expectation values of scalar fields, .... might play such a role in the breaking of symmetries.... in a theory of this type the breakdown of symmetry occurs already at the level of classical field theory...."We emphasize, that our model fits to this statement, while being valid at the same time for the Maxwell equations without any further requirements for additional space-time dimensions to keep consistency between the models.
(GrM): Green M. B., Schwarz J. H., Witten E., Superstring theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 1987
(HiP) Higgs P., Spontaneous symmetry breakdown without massless bosons, Phys. Rev. 145, 1156-1163, 1966
(KaM): Kaku M., Introduction to Superstrings and M-Theory, Springer Verlag, New York, 1988
(VeG): Veneziano, G. (1968), "Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories", Nuovo Cim. 57A, 190
(LoC): Lovelace C., "Veneziano Theory", CERN - Geneva, Ref. TH. 1123-CERN, 15 January 1996
(VeW) Velte W., Direkte Methoden der Variationsrechnung, B. G. Teubner, Stuttgart, 1976
(WeH): Weyl H., "Space, time, matter", Dover Publications, Inc., 1922
(WeH1) Weyl H., Gravitation un Elektrizität, Sitzber. Preuss. Akad. Wiss. 465-480, 1918
(YaC) Yang C. N., Mills R. L., Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96, 191-195, 1954
(BlD) Bleecker D., Gauge Theory and Variational Principles, Dover Publications, Inc., 1981