Foundations of the Theory of Dynamical Systems of Infinitely Many Degrees of Freedom, II

1961 ◽  
Vol 13 ◽  
pp. 1-18 ◽  
Author(s):  
I. E. Segal
Keyword(s):  
Degrees Of Freedom ◽  
Physical Theory ◽  
Original Work ◽  
Free Field ◽  
Space Time ◽  
The Other ◽  
Quantum Field ◽  
Bona Fide ◽  
Finite Space ◽  
Physical Result

The notion of quantum field remains at this time still rather elusive from a rigorous standpoint. In conventional physical theory such a field is defined in essentially the same way as in the original work of Heisenberg and Pauli (1) by a function ϕ(x, y, z, t) on space-time whose values are operators. It was recognized very early, however, by Bohr and Rosenfeld (2) that, even in the case of a free field, no physical meaning could be attached to the values of the field at a particular point—only the suitably smoothed averages over finite space-time regions had such a meaning. This physical result has a mathematical counterpart in the impossibility of formulating ϕ(x, y, z, t) as a bona fide operator for even the simplest fields (in any fashion satisfying the most elementary non-trivial theoretical desiderata), while on the other hand for suitable functions f, the integral ∫ϕ(x, y, zy t)f(x, y, z, t)dxdydzdt could be so formulated.

scholarly journals Atom-Field Interaction: From Vacuum Fluctuations to Quantum Radiation and Quantum Dissipation or Radiation Reaction

Physics ◽  
2019 ◽  
Vol 1 (3) ◽  
pp. 430-444 ◽  
Author(s):  
Jen-Tsung Hsiang ◽  
B. L. Hu
Keyword(s):  
Degrees Of Freedom ◽  
Mutual Influence ◽  
Equations Of Motion ◽  
Free Field ◽  
Radiation Reaction ◽  
Linear Response Theory ◽  
Quantum Field ◽  
Quantum Dissipation ◽  
Vacuum Fluctuations ◽  
Quantum Radiation

In this paper, we dwell on three issues: (1) revisit the relation between vacuum fluctuations and radiation reaction in atom-field interactions, an old issue that began in the 1970s and settled in the 1990s with its resolution recorded in monographs; (2) the fluctuation–dissipation relation (FDR) of the system, pointing out the differences between the conventional form in linear response theory (LRT) assuming ultra-weak coupling between the system and the bath, and the FDR in an equilibrated final state, relaxed from the nonequilibrium evolution of an open quantum system; (3) quantum radiation from an atom interacting with a quantum field: We begin with vacuum fluctuations in the field acting on the internal degrees of freedom (idf) of an atom, adding to its dynamics a stochastic component which engenders quantum radiation whose backreaction causes quantum dissipation in the idf of the atom. We show explicitly how different terms representing these processes appear in the equations of motion. Then, using the example of a stationary atom, we show how the absence of radiation in this simple cases is a result of complex cancellations, at a far away observation point, of the interference between emitted radiation from the atom and the local fluctuations in the free field. In so doing we point out in Issue 1 that the entity which enters into the duality relation with vacuum fluctuations is not radiation reaction, which can exist as a classical entity, but quantum dissipation. Finally, regarding issue 2, we point out for systems with many atoms, the co-existence of a set of correlation-propagation relations (CPRs) describing how the correlations between the atoms are related to the propagation of their (retarded non-Markovian) mutual influence manifesting in the quantum field. The CPR is absolutely crucial in keeping the balance of energy flows between the constituents of the system, and between the system and its environment. Without the consideration of this additional relation in tether with the FDR, dynamical self-consistency cannot be sustained. A combination of these two sets of relations forms a generalized matrix FDR relation that captures the physical essence of the interaction between an atom and a quantum field at arbitrary coupling strength.

DIMENSION OF SPACE-TIME

1986 ◽  
Vol 01 (04) ◽  
pp. 971-990 ◽  
Author(s):  
KARL SVOZIL ◽  
ANTON ZEILINGER
Keyword(s):  
Field Theory ◽  
Relativistic Quantum ◽  
Space Time ◽  
Quantum Field ◽  
Relativistic Quantum Field Theory ◽  
Real Experiment ◽  
Electromagnetic Moment ◽  
Theoretical Predictions ◽  
Finite Space ◽  
Operational Dimension

In order to make it operationally accessible, it is proposed that the notion of the dimension of space-time be based on measure-theoretic concepts, thus admitting the possibility of noninteger dimensions. It is found then, that the Hausdorff covering procedure is operationally unrealizable because of the inherent finite space-time resolution of any real experiment. We therefore propose to define an operational dimension which, due to the quantum nature of the coverings, is smaller than the idealized Hausdorff dimension. As a consequence of the dimension of space-time less than four, relativistic quantum field theory becomes finite. Also, the radiative corrections of perturbation theory are sensitive on the actual value of the dimension 4–ε. Present experimental results and standard theoretical predictions for the electromagnetic moment of the electron seem to suggest a nonvanishing value for ε.

scholarly journals ALIENS AND UFO’S

2021 ◽  
pp. 49-55
Author(s):  
Deep Bhattacharjee
Keyword(s):  
String Theory ◽  
Local Time ◽  
Degrees Of Freedom ◽  
Dimensional Regularization ◽  
Higher Dimensions ◽  
Space Time ◽  
The Other ◽  
Energy Conditions ◽  
Normal Space ◽  
Non Local

Are aliens transdimensional entities?, By this I have meant to say those beings rather living beings that are existent on different dimensions greater than our 4 that is our normal space-time dimensions. We have enough evidence from the Lorentz generators of the string theory that the maximum space-time dimensions can take the value upto 10, if time is considered as a singular dimension. However, in certain theories like the M-theory where there exists the 11th dimensions as SUGRA or supersymmetric gravity (graviton with gravitino) which on F-theory the dimensional regularization has been taken to 12 by splitting time into 2 dimensions with a higher degrees of freedom. It is very much possible and probable that in a higher dimensions d≫4 time can take the form of 2D where the local nature of 1D time would behave as a singular loop of various multiplicities as a non-local 2D element. In our 4 dimensions also, if we are sustained to believe time being a non-local element with the Einstein’s principle of “spontaneously happening past, present and future then we ought to believe that 1D local time itself acts as a 2D non-local time. Considering the dimensional range of spatial and temporal factor ~ 3+1 it can be said that there are higher ~ 6 additional spatial dimensions which are connected with the lower ~ 3+1 co-dimensions in the form of various inter-dimensional tunnels that specifies the initial and final positions from a lower dimensions to a higher dimensions. Without considering, in this paper, the respective size of the dimensions, if all those space-time dimensions ~ 6+1 are there, then it’s probable that there exists some particular creatures over those various dimensions. However, just as 4D consists of all the 3, 2, 1D’s, similarly the higher order dimensions like 10D would also contains 9, 8, 7, 6, 5…. 1D’s, therefore it can be said that 10D is very unstable and chaotic because of the intersections of various lower co-dimensions that exists in reality. There must be an inter-dimensional membrane that protects one dimensions from the other and those inter-dimensional tunnels, that exists in between (connecting) those dimensions must have an unstable mouth wandering unpredictably from one hyper-surface to another where if anyone gets caught in any of those tunnels mouth’ devoid of any singularities and horizons, then they ought to travel to the other dimensions that will always be >4 as permitted by the laws of physics standing on the 4D universe. Now, analyzing the vehicles that they used to transport from one point in spacetime to other, A detailed analysis on the engineering and phenomenology with respect to mechanisms of the unidentified aerial objects has been carried out extensively on the paper depicting why they are more advanced and on what mechanisms are they capable of the interstellar & intergalactic travel by virtue of electrohydrodynamics and semi-quantum kinetics. KEYWORDS – Hyper-surfaces; Hyper-membranes; Inter-dimensional tunnel, Cauchy horizon, Singularity, Cross-sections, Monodromy, Time slices, Co-dimensions; String theory; Average null energy conditions; f(R,T) gravity.

Retroaktive Neuverhandlung

2018 ◽  
Vol 63 (1) ◽  
Author(s):  
Daniel Martin Feige
Keyword(s):  
Original Work ◽  
The Other ◽  
Final Part ◽  
Critical Reading ◽  
The Third ◽  
Works Of Art ◽  
The Relationship

Der Beitrag widmet sich der Frage historischer Folgeverhältnisse in der Kunst. Gegenüber dem Gedanken, dass es ein ursprüngliches Werk in der Reihe von Werken gibt, das späteren Werken seinen Sinn gibt, schlägt der Text vor, das Verhältnis umgekehrt zu denken: Im Lichte späterer Werke wird der Sinn früherer Werke neu ausgehandelt. Dazu geht der Text in drei Schritten vor. Im ersten Teil formuliert er unter der Überschrift ›Form‹ in kritischer Abgrenzung zu Danto und Eco mit Adorno den Gedanken, dass Kunstwerke eigensinnig konstituierte Gegenstände sind. Die im Gedanken der Neuverhandlung früherer Werke im Lichte späterer Werke vorausgesetzte Unbestimmtheit des Sinns von Kunstwerken wird im zweiten Teil unter dem Schlagwort ›Zeitlichkeit‹ anhand des Paradigmas der Improvisation erörtert. Der dritte und letzte Teil wendet diese improvisatorische Logik unter dem Label ›Neuaushandlung‹ dann dezidiert auf das Verhältnis von Vorbild und Nachbild an. The article proposes a new understanding of historical succession in the realm of art. In contrast to the idea that there is an original work in the series of works that gives meaning to the works that come later, the text proposes to think it exactly the other way round: in the light of later works, the meanings of earlier works are renegotiated. The text proceeds in three steps to develop this idea. Under the heading ›Form‹ it develops in the first part a critical reading of Danto’s and Eco’s notion of the constitution of the artworks and argues with Adorno that each powerful work develops its own language. In the second part, the vagueness of the meaning of works of art presupposed in the idea of renegotiating earlier works in the light of later works is discussed under the term ›Temporality‹ in terms of the logic of improvisation. The third and final part uses this improvisational logic under the label ›Renegotiation‹ to understand the relationship between model and afterimage in the realm of art.

scholarly journals GENERALIZED W3-STRINGS FROM FREE FIELDS

1995 ◽  
Vol 10 (06) ◽  
pp. 515-524 ◽  
Author(s):  
J. M. FIGUEROA-O'FARRILL ◽  
C. M. HULL ◽  
L. PALACIOS ◽  
E. RAMOS
Keyword(s):  
Bosonic Strings ◽  
Free Field ◽  
Space Time ◽  
General Construction ◽  
Special Cases ◽  
Free Fields ◽  
Rule Out ◽  
General Conditions ◽  
String Theories

The conventional quantization of w3-strings gives theories which are equivalent to special cases of bosonic strings. We explore whether a more general quantization can lead to new generalized W3-string theories by seeking to construct quantum BRST charges directly without requiring the existence of a quantum W3-algebra. We study W3-like strings with a direct space-time interpretation — that is, with matter given by explicit free field realizations. Special emphasis is placed on the attempt to construct a quantum W-string associated with the magic realizations of the classical w3-algebra. We give the general conditions for the existence of W3-like strings, and comment on how the known results fit into our general construction. Our results are negative: we find no new consistent string theories, and in particular rule out the possibility of critical strings based on the magic realizations.

scholarly journals How low can vacuum energy go when your fields are finite-dimensional?

2019 ◽  
Vol 28 (14) ◽  
pp. 1944006
Author(s):  
ChunJun Cao ◽  
Aidan Chatwin-Davies ◽  
Ashmeet Singh
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Degrees Of Freedom ◽  
Vacuum Energy ◽  
Quantum Field ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Finite Dimensionality ◽  
Klein Gordon ◽  
Dimensional Version

According to the holographic bound, there is only a finite density of degrees of freedom in space when gravity is taken into account. Conventional quantum field theory does not conform to this bound, since in this framework, infinitely many degrees of freedom may be localized to any given region of space. In this paper, we explore the viewpoint that quantum field theory may emerge from an underlying theory that is locally finite-dimensional, and we construct a locally finite-dimensional version of a Klein–Gordon scalar field using generalized Clifford algebras. Demanding that the finite-dimensional field operators obey a suitable version of the canonical commutation relations makes this construction essentially unique. We then find that enforcing local finite dimensionality in a holographically consistent way leads to a huge suppression of the quantum contribution to vacuum energy, to the point that the theoretical prediction becomes plausibly consistent with observations.

CRYSTALLINE GRAVITY

2009 ◽  
Vol 24 (18n19) ◽  
pp. 3243-3255 ◽  
Author(s):  
GERARD 't HOOFT
Keyword(s):  
Finite Number ◽  
Lorentz Group ◽  
Degrees Of Freedom ◽  
Holonomy Group ◽  
Discrete Subgroup ◽  
Analytic Solutions ◽  
Space Time ◽  
Exact Analytic Solutions ◽  
Finite Domains ◽  
Dimensional Crystal

Matter interacting classically with gravity in 3+1 dimensions usually gives rise to a continuum of degrees of freedom, so that, in any attempt to quantize the theory, ultraviolet divergences are nearly inevitable. Here, we investigate a theory that only displays a finite number of degrees of freedom in compact sections of space-time. In finite domains, one has only exact, analytic solutions. This is achieved by limiting ourselves to straight pieces of string, surrounded by locally flat sections of space-time. Next, we suggest replacing in the string holonomy group, the Lorentz group by a discrete subgroup, which turns space-time into a 4-dimensional crystal with defects.

scholarly journals ε-EXPANSION IN QUANTUM FIELD THEORY IN CURVED SPACE–TIME

1998 ◽  
Vol 13 (16) ◽  
pp. 2857-2874
Author(s):  
IVER H. BREVIK ◽  
HERNÁN OCAMPO ◽  
SERGEI ODINTSOV
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Space Time ◽  
Quantum Field ◽  
Curved Space ◽  
Njl Model ◽  
Effective Lagrangian ◽  
Special Solutions ◽  
Curvature Approximation ◽  
Symmetric Phase

We discuss ε-expansion in curved space–time for asymptotically free and asymptotically nonfree theories. The existence of stable and unstable fixed points is investigated for fϕ4 theory and SU(2) gauge theory. It is shown that ε-expansion maybe compatible with aysmptotic freedom on special solutions of the RG equations in a special ase (supersymmetric theory). Using ε-expansion RG technique, the effective Lagrangian for covariantly constant gauge SU(2) field and effective potential for gauged NJL model are found in (4-ε)-dimensional curved space (in linear curvature approximation). The curvature-induced phase transitions from symmetric phase to asymmetric phase (chromomagnetic vacuum and chiral symmetry broken phase, respectively) are discussed for the above two models.

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