EJTP ElEctronic Journal of thEorEtical Physics
Volume 14 April, 2018
Number 37
http://www.ejtp.com E-mail:[email protected]
editorsIgnazio LicataAmmar Sakaji
Aracne editrice
Copyright © MMXVIIGioacchino Onorati editore S.r.l. – unipersonale
via Vittorio Veneto, 2000020 Canterano (RM)
(06) 45551463
isbn 978–88–255–1599–2
No part of this book may be reproduced by print, photoprint, microfilm, microfiche, or any other means,
without publisher’s authorization.
I edizione: June 2018
Table of Contents
1. In memory of Stephen Hawking............................................................... i I. LIcata
2. Preface...................................................................................................... v a. SakajI
3. Projective Limits of State Spaces: Quantum Field Theory Without a Va-cuum......................................................................................................... 1
S. Lanery
4. Noncommutative Structure of Massive Bosonic Strings............................ 21 n. ManSour, e. DIaf, M.B. SeDra
5. Minimal Length, Minimal Inverse Temperature, Measurability and Black Hole........................................................................................................... 35
a.e. ShaLyt–MargoLIn
6. Poisson Bracket and Symplectic Structure of Covariant Canonical Forma- lism of Fields............................................................................................ 55
y. kaMInaga
7. Neutrino Masses and Effective Majorana Mass from a Cobimaximal Neu- trino Mixing Matrix.................................................................................... 73
a. DaManIk
8. Relativistic Klein–Gordan Equation with Position Dependent Mass for q–deformed Modifed Eckart plus Hylleraas potential............................... 79
S. Sur, S. DeBnath
9. Investigation Fermionic Quantum Walk for Detecting Nonisomorph Cospe- ctral Graphs............................................................................................... 91
M.a. jafarIzaDeh, f. eghBaLIfaMy, S. naMIz
10. Thermodynamics of Hot Quantum Scalar Field in a (D + 1) Dimensional Curved Spacetime..................................................................................... 115
W.a. rojaS, j.r. arenaS S.
11. Spin and Zitterbewegung in a Field Theory of the Electron...................... 125 e. recaMI, g. SaLeSI
12. Solutions to the Gravitational Field Equations in Curved Phase–Spaces. 145 c. caStro
13. Electromagnetic Media in pp–wave Spacetime...................................... 161 M. fathI
14. Validation of the Hadron Mass Quantization from Experimental Hadronic Regge Trajectories.................................................................................... 179
n. hothI, S. BISht
15. Neimark–Sacker and Closed Invariant Curve Bifurcations of A Two Dimensional Map Used For Cryptography............................................... 195
y. yahIaouI, n. akrouney
16. Physics of Currents and Potentials IV. Dirac Space and Dirac Vectors in the Quantum Relativistic Theory............................................................ 213
V.a. teMneko
Editor in Chief
Ignazio Licata
Foundations of Quantum Mechanics, Complex System & Computation in Physics and Biology, IxtuCyber for Complex Systems , and ISEM, Institute for Scientific Methodology, Palermo, Sicily – Italy
editor[AT]ejtp.info Email: ignazio.licata[AT]ejtp.info
ignazio.licata[AT]ixtucyber.org
Co-Editor
Ammar Sakaji
Theoretical Condensed Matter, Mathematical Physics Center of Theoretical Physics and Astrophysics, Amman-Jordan Tel:+962778195003 Email: [email protected]
Editorial Board
Gerardo F. Torres del Castillo
Mathematical Physics, Classical Mechanics, General Relativity, Universidad Autónoma de Puebla, México, Email:gtorres[AT]fcfm.buap.mx Torresdelcastillo[AT]gmail.com
Leonardo Chiatti
Medical Physics Laboratory AUSL VT Via Enrico Fermi 15, 01100 Viterbo (Italy) Tel : (0039) 0761 1711055 Fax (0039) 0761 1711055 Email: fisica1.san[AT]asl.vt.it chiatti[AT]ejtp.info
Editor in Chief
Ignazio Licata
Foundations of Quantum Mechanics, Complex System & Computation in Physics and Biology, IxtuCyber for Complex Systems , and ISEM, Institute for Scientific Methodology, Palermo, Sicily – Italy
editor[AT]ejtp.info Email: ignazio.licata[AT]ejtp.info
ignazio.licata[AT]ixtucyber.org
Co-Editor
Ammar Sakaji
Theoretical Condensed Matter, Mathematical Physics Center of Theoretical Physics and Astrophysics, Amman-Jordan Tel:+962778195003 Email: [email protected]
Editorial Board
Gerardo F. Torres del Castillo
Mathematical Physics, Classical Mechanics, General Relativity, Universidad Autónoma de Puebla, México, Email:gtorres[AT]fcfm.buap.mx Torresdelcastillo[AT]gmail.com
Leonardo Chiatti
Medical Physics Laboratory AUSL VT Via Enrico Fermi 15, 01100 Viterbo (Italy) Tel : (0039) 0761 1711055 Fax (0039) 0761 1711055 Email: fisica1.san[AT]asl.vt.it chiatti[AT]ejtp.info
Francisco Javier Chinea
Differential Geometry & General Relativity, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Spain, E-mail: chinea[AT]fis.ucm.es
Maurizio Consoli
Non Perturbative Description of Spontaneous Symmetry Breaking as a Condensation Phenomenon, Emerging Gravity and Higgs Mechanism, Dip. Phys., Univ. CT, INFN,Italy
Email: Maurizio.Consoli[AT]ct.infn.it
Avshalom Elitzur
Foundations of Quantum Physics ISEM, Institute for Scientific Methodology, Palermo, Italy
Email: Avshalom.Elitzur[AT]ejtp.info
Sharmanthie Fernando
General Theory of Relativity and Black Holes Department of Physics, Geology and Engineering Technology, Northern Kentucky University, KY 41099, USA
Email: Fernando[AT]nku.edu
Elvira Fortunato
Quantum Devices and Nanotechnology:
Departamento de Ciência dos Materiais CENIMAT, Centro de Investigação de Materiais I3N, Instituto de Nanoestruturas, Nanomodelação e Nanofabricação FCT-UNL Campus de Caparica 2829-516 Caparica Portugal Tel: +351 212948562; Directo:+351 212949630 Fax: +351 212948558 Email:emf[AT]fct.unl.pt elvira.fortunato[AT]fct.unl.pt
Tepper L. Gill
Mathematical Physics, Quantum Field Theory Department of Electrical and Computer Engineering Howard University, Washington, DC, USA
Email: tgill[AT]Howard.edu tgill[AT]ejtp.info
Alessandro Giuliani
Mathematical Models for Molecular Biology Senior Scientist at Istituto Superiore di Sanità Roma-Italy
Email: alessandro.giuliani[AT]iss.it
Vitiello Giuseppe
Quantum Field Theories, Neutrino Oscillations, Biological Systems Dipartimento di Fisica Università di Salerno Baronissi (SA) - 84081 Italy Phone: +39 (0)89 965311 Fax : +39 (0)89 953804 Email: [email protected]
Richard Hammond
General Relativity High energy laser interactions with charged particles Classical equation of motion with radiation reaction Electromagnetic radiation reaction forces Department of Physics University of North Carolina at Chapel Hill, USA Email: rhammond[AT]email.unc.edu
Arbab Ibrahim
Theoretical Astrophysics and Cosmology Department of Physics, Faculty of Science, University of Khartoum, P.O. Box 321, Khartoum 11115, Sudan
Email: aiarbab[AT]uofk.edu arbab_ibrahim[AT]ejtp.info
Kirsty Kitto
Quantum Theory and Complexity Information Systems | Faculty of Science and Technology Queensland University of Technology Brisbane 4001 Australia
Email: kirsty.kitto[AT]qut.edu.au
Hagen Kleinert
Quantum Field Theory Institut für Theoretische Physik, Freie Universit¨at Berlin, 14195 Berlin, Germany
Email: h.k[AT]fu-berlin.de
Francisco Javier Chinea
Differential Geometry & General Relativity, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Spain, E-mail: chinea[AT]fis.ucm.es
Maurizio Consoli
Non Perturbative Description of Spontaneous Symmetry Breaking as a Condensation Phenomenon, Emerging Gravity and Higgs Mechanism, Dip. Phys., Univ. CT, INFN,Italy
Email: Maurizio.Consoli[AT]ct.infn.it
Avshalom Elitzur
Foundations of Quantum Physics ISEM, Institute for Scientific Methodology, Palermo, Italy
Email: Avshalom.Elitzur[AT]ejtp.info
Sharmanthie Fernando
General Theory of Relativity and Black Holes Department of Physics, Geology and Engineering Technology, Northern Kentucky University, KY 41099, USA
Email: Fernando[AT]nku.edu
Elvira Fortunato
Quantum Devices and Nanotechnology:
Departamento de Ciência dos Materiais CENIMAT, Centro de Investigação de Materiais I3N, Instituto de Nanoestruturas, Nanomodelação e Nanofabricação FCT-UNL Campus de Caparica 2829-516 Caparica Portugal Tel: +351 212948562; Directo:+351 212949630 Fax: +351 212948558 Email:emf[AT]fct.unl.pt elvira.fortunato[AT]fct.unl.pt
Tepper L. Gill
Mathematical Physics, Quantum Field Theory Department of Electrical and Computer Engineering Howard University, Washington, DC, USA
Email: tgill[AT]Howard.edu tgill[AT]ejtp.info
Alessandro Giuliani
Mathematical Models for Molecular Biology Senior Scientist at Istituto Superiore di Sanità Roma-Italy
Email: alessandro.giuliani[AT]iss.it
Vitiello Giuseppe
Quantum Field Theories, Neutrino Oscillations, Biological Systems Dipartimento di Fisica Università di Salerno Baronissi (SA) - 84081 Italy Phone: +39 (0)89 965311 Fax : +39 (0)89 953804 Email: [email protected]
Richard Hammond
General Relativity High energy laser interactions with charged particles Classical equation of motion with radiation reaction Electromagnetic radiation reaction forces Department of Physics University of North Carolina at Chapel Hill, USA Email: rhammond[AT]email.unc.edu
Arbab Ibrahim
Theoretical Astrophysics and Cosmology Department of Physics, Faculty of Science, University of Khartoum, P.O. Box 321, Khartoum 11115, Sudan
Email: aiarbab[AT]uofk.edu arbab_ibrahim[AT]ejtp.info
Kirsty Kitto
Quantum Theory and Complexity Information Systems | Faculty of Science and Technology Queensland University of Technology Brisbane 4001 Australia
Email: kirsty.kitto[AT]qut.edu.au
Hagen Kleinert
Quantum Field Theory Institut für Theoretische Physik, Freie Universit¨at Berlin, 14195 Berlin, Germany
Email: h.k[AT]fu-berlin.de
Wai-ning Mei
Condensed matter Theory Physics Department University of Nebraska at Omaha,
Omaha, Nebraska, USA Email: wmei[AT]mail.unomaha.edu physmei[AT]unomaha.edu
Beny Neta
Applied Mathematics Department of Mathematics Naval Postgraduate School 1141 Cunningham Road Monterey, CA 93943, USA Email: byneta[AT]gmail.com
Peter O'Donnell
General Relativity & Mathematical Physics, Homerton College, University of Cambridge, Hills Road, Cambridge CB2 8PH, UK E-mail: po242[AT]cam.ac.uk
Rajeev Kumar Puri
Theoretical Nuclear Physics, Physics Department, Panjab University Chandigarh -160014, India Email: drrkpuri[AT]gmail.com rkpuri[AT]pu.ac.in
Haret C. Rosu
Advanced Materials Division Institute for Scientific and Technological Research (IPICyT) Camino a la Presa San José 2055 Col. Lomas 4a. sección, C.P. 78216 San Luis Potosí, San Luis Potosí, México Email: hcr[AT]titan.ipicyt.edu.mx
Zdenek Stuchlik
Relativistic Astrophysics Department of Physics, Faculty of Philosophy and Science, Silesian University, Bezru covo n´am. 13, 746 01 Opava, Czech Republic Email: Zdenek.Stuchlik[AT]fpf.slu.cz
Fabrizio Tamburini
Electromagnetic Vorticity, General Relativity, Quantum mechanics, Astrophysics of compact objects Scientist in Residence at ZKM - Zentrum für Kunst und Medientechnologie Lorenzstraße 19, 76135 Karlsruhe, Germany www.zkm.de/ Email: fabrizio.tamburini[AT]gmail.com
S.I. Themelis
Atomic, Molecular & Optical Physics Foundation for Research and Technology - Hellas P.O. Box 1527, GR-711 10 Heraklion, Greece Email: stheme[AT]iesl.forth.gr
Yurij Yaremko
Special and General Relativity, Electrodynamics of classical charged particles, Mathematical Physics, Institute for Condensed Matter Physics of Ukrainian National Academy of Sciences 79011 Lviv, Svientsytskii Str. 1 Ukraine Email: yu.yaremko[AT]gmail.com yar[AT]icmp.lviv.ua
yar[AT]ph.icmp.lviv.ua
Nicola Yordanov
Physical Chemistry Bulgarian Academy of Sciences, BG-1113 Sofia, Bulgaria Telephone: (+359 2) 724917 , (+359 2) 9792546
Email: ndyepr[AT]ic.bas.bg ndyepr[AT]bas.bg
Former Editors:
Ignazio Licata, Editor in Chief (August 2015-)
Ignazio Licata, Editor in Chief (October 2009- August 2012)
Losé Luis López-Bonilla, Co-Editor (2008-2012)
Ammar Sakaji, Founder and Editor in Chief (2003- October 2009) and (August 2012- August2015).
Wai-ning Mei
Condensed matter Theory Physics Department University of Nebraska at Omaha,
Omaha, Nebraska, USA Email: wmei[AT]mail.unomaha.edu physmei[AT]unomaha.edu
Beny Neta
Applied Mathematics Department of Mathematics Naval Postgraduate School 1141 Cunningham Road Monterey, CA 93943, USA Email: byneta[AT]gmail.com
Peter O'Donnell
General Relativity & Mathematical Physics, Homerton College, University of Cambridge, Hills Road, Cambridge CB2 8PH, UK E-mail: po242[AT]cam.ac.uk
Rajeev Kumar Puri
Theoretical Nuclear Physics, Physics Department, Panjab University Chandigarh -160014, India Email: drrkpuri[AT]gmail.com rkpuri[AT]pu.ac.in
Haret C. Rosu
Advanced Materials Division Institute for Scientific and Technological Research (IPICyT) Camino a la Presa San José 2055 Col. Lomas 4a. sección, C.P. 78216 San Luis Potosí, San Luis Potosí, México Email: hcr[AT]titan.ipicyt.edu.mx
Zdenek Stuchlik
Relativistic Astrophysics Department of Physics, Faculty of Philosophy and Science, Silesian University, Bezru covo n´am. 13, 746 01 Opava, Czech Republic Email: Zdenek.Stuchlik[AT]fpf.slu.cz
Fabrizio Tamburini
Electromagnetic Vorticity, General Relativity, Quantum mechanics, Astrophysics of compact objects Scientist in Residence at ZKM - Zentrum für Kunst und Medientechnologie Lorenzstraße 19, 76135 Karlsruhe, Germany www.zkm.de/ Email: fabrizio.tamburini[AT]gmail.com
S.I. Themelis
Atomic, Molecular & Optical Physics Foundation for Research and Technology - Hellas P.O. Box 1527, GR-711 10 Heraklion, Greece Email: stheme[AT]iesl.forth.gr
Yurij Yaremko
Special and General Relativity, Electrodynamics of classical charged particles, Mathematical Physics, Institute for Condensed Matter Physics of Ukrainian National Academy of Sciences 79011 Lviv, Svientsytskii Str. 1 Ukraine Email: yu.yaremko[AT]gmail.com yar[AT]icmp.lviv.ua
yar[AT]ph.icmp.lviv.ua
Nicola Yordanov
Physical Chemistry Bulgarian Academy of Sciences, BG-1113 Sofia, Bulgaria Telephone: (+359 2) 724917 , (+359 2) 9792546
Email: ndyepr[AT]ic.bas.bg ndyepr[AT]bas.bg
Former Editors:
Ignazio Licata, Editor in Chief (August 2015-)
Ignazio Licata, Editor in Chief (October 2009- August 2012)
Losé Luis López-Bonilla, Co-Editor (2008-2012)
Ammar Sakaji, Founder and Editor in Chief (2003- October 2009) and (August 2012- August2015).
Electronic Journal of Theoretical Physics 14, No. 37 (2018) i
EJTP V14, No 37
He Lived Here
In memory of Stephen Hawking
Oxford, 8 January 1942 – Cambridge, 14 March 2018
What a strange awakening today. Stephen Hawking – escaped from an infaust diagnosis
21 years ago and from many fatal surgeries- is gone.
My generation has grown up with catchphrases such as “where were you when John
Lennon died?” I think that Stephen Hawking death will bring back similar questions in
the future. As it has been said for Einstein: “He lived here”. I’m not speaking only of
the powerful and empathetic relation he and his work had with the media and audience.
All of us admired Eddie Redmayne in The Theory of Everything and, earlier, Benedict
Cumberbutch playing the role of Hawking, all of us know something about black holes
and their radiation; and A Brief History of Time is surely one of the the most successful
book of all time.
Actually, there another reason why the Stephen Hawking death will be stuck in our minds.
Just for once, the image in the media was really the man behind and beyond the news.
You could always perceive he was a man of stature and an intense person, there was
something unique between the brightening of his eyes and the lines of his most technical
papers. Einstein used to say that a theoretical physicist can appear to be an opportunist
with no scruples to epistemologists. The reason lies in the fact that a scientist uses precise
tools, mathematics for theoretical physicists, and not the power of interpretations; in no
way a scientist worries about giving a frame to a result so mimicking a philosopher.
That’s where Hawking was, maybe, the most secular among the scientists. He never
married a theory, but he wooed them all, just like he never failed to smile at a pretty
woman. He always questioned how far a theory could we lead, and we could actually
say about the Universe. Among all the things that gave him everlasting fame there are
Fig. 1 Stephen Hawking with W.J. Kaufmann (1977)
Electronic Journal of Theoretical Physics 14, No. 37 (2018) i
EJTP V14, No 37
He Lived Here
In memory of Stephen Hawking
Oxford, 8 January 1942 – Cambridge, 14 March 2018
What a strange awakening today. Stephen Hawking – escaped from an infaust diagnosis
21 years ago and from many fatal surgeries- is gone.
My generation has grown up with catchphrases such as “where were you when John
Lennon died?” I think that Stephen Hawking death will bring back similar questions in
the future. As it has been said for Einstein: “He lived here”. I’m not speaking only of
the powerful and empathetic relation he and his work had with the media and audience.
All of us admired Eddie Redmayne in The Theory of Everything and, earlier, Benedict
Cumberbutch playing the role of Hawking, all of us know something about black holes
and their radiation; and A Brief History of Time is surely one of the the most successful
book of all time.
Actually, there another reason why the Stephen Hawking death will be stuck in our minds.
Just for once, the image in the media was really the man behind and beyond the news.
You could always perceive he was a man of stature and an intense person, there was
something unique between the brightening of his eyes and the lines of his most technical
papers. Einstein used to say that a theoretical physicist can appear to be an opportunist
with no scruples to epistemologists. The reason lies in the fact that a scientist uses precise
tools, mathematics for theoretical physicists, and not the power of interpretations; in no
way a scientist worries about giving a frame to a result so mimicking a philosopher.
That’s where Hawking was, maybe, the most secular among the scientists. He never
married a theory, but he wooed them all, just like he never failed to smile at a pretty
woman. He always questioned how far a theory could we lead, and we could actually
say about the Universe. Among all the things that gave him everlasting fame there are
Fig. 1 Stephen Hawking with W.J. Kaufmann (1977)
ii Electronic Journal of Theoretical Physics 14, No. 37 (2018)
two problems which really stay at the extreme borders of knowledge. One deals with
the final stage of massive stars, the famous Black Holes, which, according to Einstein,
during the last stage of their lives should enter on infinite collapse, a singularity. Laplace
had already saw it, as well as Oppenheimer and Landau later, up to Wheeler and his
master Sciama. Nobody had ever investigated it before as Hawking did. Singularity
had wandered like a monster in specialized reviews and journals for some years, later –
thanks to Hawking and Penrose – it became clear that it was just a structural limitation
of Einstein gravitation theory and it was time to give room to a new theory, the quantum
gravity, which is still a frontline topic in theoretical physics. Stephen Hawking was the
one who reached some milestones in this new field, the black hole radiation and the
information analysis of a physical system with Hawking-Bekenstein formula. Black holes
were just an exercise, because now Stephen was ready to look at the more mysterious
singularity, the Big Bang. It was about in the ’80s, he and Jim Hartle proposed the
no-boundary Universe, a charming expression that we can coarsley translate by saying
that space and time emerge from a quantum nebulosity ; something similar to the Nicola
Cusano Universe, where there is no before and no after, where each point is the center.
Or, just to be a bit more technical, where time is curved and imaginary before collapsing
into what we see and what the Standard Model describes.
Similarly to all the other physicists, I happen to quote Stephen thousand times, and
every time it was an occasion to read his works again. I admired his ability to build
an apparently impenetrable castle of mathematics all around a strong idea. He could
have been an excellent chess player. I say “apparently” because Hawking knew very
well that mathematics was a sublime form of rethoric which could always be attacked or
taken apart. Or started from scratch. What really makes the difference for a physical
theory are generalities and the steadiness of its starting points. Sometimes, a weak point
could be found in Stephen’s approach (shrewd, very subtle! ), but , at the same time,
you couldn’t help but notice how the question had been posed with absolute clearness
and how it would be really difficult to do it better. Leonardo Chiatti and I started from
Hartle-Hawking theory to develop the idea of the Archaic Universe [1,2,3 ]and, recently,
Fabrizio Tamburini, Maria Felicia de Laurentis and I have discovered a particular mode of
Hawking radiation, the so-called soft hairs. There’s only a case when Hawking admitted
to be defeated, in front of a young Don Page, about the end of the Universe, namely
about the possibility that the whole wave-function rewinds to go back to origins. Like it
happened some years before with Kip Thorne about the possibility to discover a black
hole in Cygnus X-1, also in that occasion a stake was paid: a magazine subscription (in
the case of Kip, it was a yearly subscription to Playboy). In my opinion, the Hawking
idea is well-grounded, so the last word has not been spoken.
Maybe, the most don’t know that there is a beautiful theatrical play titled God and
Stephen Hawking on Stephen Hawking life and his struggles with his disease and the
biggest mysteries of the Universe. The author, Robin Hawdon, was really within “the
zone” when wrote it, you can find in the play the same humor which has became the
irreducible trait of Stephen.
Electronic Journal of Theoretical Physics 14, No. 37 (2018) iii
In the end there are some cues echoing the closing lines of A Brief History of Time, where
Stephen reaffirms his faith in a Final Reason, it equates him with giants like Einstein.
Let’s listen to it once again:
STEPHEN: I do know it is there, inherent to the infinite experiment of the Universe. A
solution that – differently from any metaphysical theory and belief – will look to be so
clear....so patent...and we will realize it has been with us all the time.
Bye Stephen!
Ignazio Licata
[1] Ignazio Licata: Universe Without Singularities. A Group Approach to de Sitter
Cosmology, EJTP, vol. 3 nr. 10 (2006), pp. 211-224
[2] Ignazio Licata, Leonardo Chiatti: The Archaic Universe: Big Bang, Cosmological
Term and the Quantum Origin of Time in Projective Cosmology, International Journal
of Theoretical Physics, vol. 48, nr. 4 (2009), pp. 1003-1018
[3] Ignazio Licata, Leonardo Chiatti: Archaic Universe and Cosmological Model: ”Big-
Bang” as Nucleation by Vacuum, International Journal of Theoretical Physics, vol. 49,
nr. 10, (2010) pp. 2379-2402
[4] Fabrizio Tamburini, Mariafelicia De Laurentis, Ignazio Licata and Bo Thide Twisted
Soft Photon Hair Implants on Black Holes, Entropy (2017), 19 (9), 458
Ignazio Licata
ii Electronic Journal of Theoretical Physics 14, No. 37 (2018)
two problems which really stay at the extreme borders of knowledge. One deals with
the final stage of massive stars, the famous Black Holes, which, according to Einstein,
during the last stage of their lives should enter on infinite collapse, a singularity. Laplace
had already saw it, as well as Oppenheimer and Landau later, up to Wheeler and his
master Sciama. Nobody had ever investigated it before as Hawking did. Singularity
had wandered like a monster in specialized reviews and journals for some years, later –
thanks to Hawking and Penrose – it became clear that it was just a structural limitation
of Einstein gravitation theory and it was time to give room to a new theory, the quantum
gravity, which is still a frontline topic in theoretical physics. Stephen Hawking was the
one who reached some milestones in this new field, the black hole radiation and the
information analysis of a physical system with Hawking-Bekenstein formula. Black holes
were just an exercise, because now Stephen was ready to look at the more mysterious
singularity, the Big Bang. It was about in the ’80s, he and Jim Hartle proposed the
no-boundary Universe, a charming expression that we can coarsley translate by saying
that space and time emerge from a quantum nebulosity ; something similar to the Nicola
Cusano Universe, where there is no before and no after, where each point is the center.
Or, just to be a bit more technical, where time is curved and imaginary before collapsing
into what we see and what the Standard Model describes.
Similarly to all the other physicists, I happen to quote Stephen thousand times, and
every time it was an occasion to read his works again. I admired his ability to build
an apparently impenetrable castle of mathematics all around a strong idea. He could
have been an excellent chess player. I say “apparently” because Hawking knew very
well that mathematics was a sublime form of rethoric which could always be attacked or
taken apart. Or started from scratch. What really makes the difference for a physical
theory are generalities and the steadiness of its starting points. Sometimes, a weak point
could be found in Stephen’s approach (shrewd, very subtle! ), but , at the same time,
you couldn’t help but notice how the question had been posed with absolute clearness
and how it would be really difficult to do it better. Leonardo Chiatti and I started from
Hartle-Hawking theory to develop the idea of the Archaic Universe [1,2,3 ]and, recently,
Fabrizio Tamburini, Maria Felicia de Laurentis and I have discovered a particular mode of
Hawking radiation, the so-called soft hairs. There’s only a case when Hawking admitted
to be defeated, in front of a young Don Page, about the end of the Universe, namely
about the possibility that the whole wave-function rewinds to go back to origins. Like it
happened some years before with Kip Thorne about the possibility to discover a black
hole in Cygnus X-1, also in that occasion a stake was paid: a magazine subscription (in
the case of Kip, it was a yearly subscription to Playboy). In my opinion, the Hawking
idea is well-grounded, so the last word has not been spoken.
Maybe, the most don’t know that there is a beautiful theatrical play titled God and
Stephen Hawking on Stephen Hawking life and his struggles with his disease and the
biggest mysteries of the Universe. The author, Robin Hawdon, was really within “the
zone” when wrote it, you can find in the play the same humor which has became the
irreducible trait of Stephen.
Electronic Journal of Theoretical Physics 14, No. 37 (2018) iii
In the end there are some cues echoing the closing lines of A Brief History of Time, where
Stephen reaffirms his faith in a Final Reason, it equates him with giants like Einstein.
Let’s listen to it once again:
STEPHEN: I do know it is there, inherent to the infinite experiment of the Universe. A
solution that – differently from any metaphysical theory and belief – will look to be so
clear....so patent...and we will realize it has been with us all the time.
Bye Stephen!
Ignazio Licata
[1] Ignazio Licata: Universe Without Singularities. A Group Approach to de Sitter
Cosmology, EJTP, vol. 3 nr. 10 (2006), pp. 211-224
[2] Ignazio Licata, Leonardo Chiatti: The Archaic Universe: Big Bang, Cosmological
Term and the Quantum Origin of Time in Projective Cosmology, International Journal
of Theoretical Physics, vol. 48, nr. 4 (2009), pp. 1003-1018
[3] Ignazio Licata, Leonardo Chiatti: Archaic Universe and Cosmological Model: ”Big-
Bang” as Nucleation by Vacuum, International Journal of Theoretical Physics, vol. 49,
nr. 10, (2010) pp. 2379-2402
[4] Fabrizio Tamburini, Mariafelicia De Laurentis, Ignazio Licata and Bo Thide Twisted
Soft Photon Hair Implants on Black Holes, Entropy (2017), 19 (9), 458
Ignazio Licata
Preface
In the first quarter of 2018, we present a collection of fourteen manuscripts covering
important topics of theoretical and mathematical physics ranging from quantum walk,
gravitational waves, string theory, gauge field theories and canonical formalism, gravi-
tational thermodynamics and quantum gravity, neutrino masses and effective Majorana,
relativistic Klein-Gordan equation, thermodynamics of hot Quantum scalar field, Spin
and Zitterbewegung, solutions to the gravitational field equations in curved phase-spaces,
hadron mass quantization, Neimark-Sacker bifurcation and chaotic attractors for discrete
dynamical systems, and Dirac space in the Quantum relativistic theory.
Lanery on his paper presents a self-contained introduction of the projective limits of
state spaces: quantum field theory without a vacuum and its relations to other QFT
approaches. Mansour et al. addresses in his paper the Faddeev-Jackiw quantization
methodology in the noncommutative structure of massive Bosonic strings. Margolin
defined the gravitational thermodynamics for minimal length and minimal inverse tem-
perature. Kaminaga in paper propose the Poisson bracket for a new canonical theory.
Damanik in his work derives a neutrino mass matrix from cobimaximal neutrino mixing
matrix in parallel with effective Majorana mass. Debnath address relativistic Klein-
Gordan equation for q-deformed modified Eckart plus Hylleraas potential. Jafarizadeh
et al. on their work on graph isomorphism problem investigate Fermionic quantum walk
for detecting Nonisomorph Cospectral Graphs. Rojas et al. use the brick wall model to
calculate of free energy of quantum scalar field in a curved spacetime (D +1) dimensions.
Recami et al. in his paper ”Spin and Zitterbewegung” address the classical theory of
the electron in parallel with quantum analogue to extend a new non-linear Dirac-like
equation. Castro in his paper gives mathematical solutions to the gravitational field
equations in curved phase-spaces. Fathi presents dialectic transformation media within
gravitational waves. Hothi et al. show the validation of the Hadron mass quantization
from experimental Hadronic Regge trajectories. Yahiaoui et al. in their cryptographic
work discuss the dynamics and bifurcations of a family of two-dimensional noninvertible
maps. Temnenko in his 4th paper of the series of physics of currents and potentials
addresses Dirac space and vectors.
I want to express my sincere gratitude to the my friend Ignazio Licata for the valu-
able discussions, reviewing and excellent editorial work, and thanks to my friend Hanna
Sabat from the center of theoretical physics and astrophysics for his help in editing the
manuscripts, many thanks to our referees for their valuable comments and notes. We
thank all authors who contributed their articles for this issue.
Ammar Sakaji
Electronic Journal of Theoretical Physics 14, No. 37 (2018) v
Preface
In the first quarter of 2018, we present a collection of fourteen manuscripts covering
important topics of theoretical and mathematical physics ranging from quantum walk,
gravitational waves, string theory, gauge field theories and canonical formalism, gravi-
tational thermodynamics and quantum gravity, neutrino masses and effective Majorana,
relativistic Klein-Gordan equation, thermodynamics of hot Quantum scalar field, Spin
and Zitterbewegung, solutions to the gravitational field equations in curved phase-spaces,
hadron mass quantization, Neimark-Sacker bifurcation and chaotic attractors for discrete
dynamical systems, and Dirac space in the Quantum relativistic theory.
Lanery on his paper presents a self-contained introduction of the projective limits of
state spaces: quantum field theory without a vacuum and its relations to other QFT
approaches. Mansour et al. addresses in his paper the Faddeev-Jackiw quantization
methodology in the noncommutative structure of massive Bosonic strings. Margolin
defined the gravitational thermodynamics for minimal length and minimal inverse tem-
perature. Kaminaga in paper propose the Poisson bracket for a new canonical theory.
Damanik in his work derives a neutrino mass matrix from cobimaximal neutrino mixing
matrix in parallel with effective Majorana mass. Debnath address relativistic Klein-
Gordan equation for q-deformed modified Eckart plus Hylleraas potential. Jafarizadeh
et al. on their work on graph isomorphism problem investigate Fermionic quantum walk
for detecting Nonisomorph Cospectral Graphs. Rojas et al. use the brick wall model to
calculate of free energy of quantum scalar field in a curved spacetime (D +1) dimensions.
Recami et al. in his paper ”Spin and Zitterbewegung” address the classical theory of
the electron in parallel with quantum analogue to extend a new non-linear Dirac-like
equation. Castro in his paper gives mathematical solutions to the gravitational field
equations in curved phase-spaces. Fathi presents dialectic transformation media within
gravitational waves. Hothi et al. show the validation of the Hadron mass quantization
from experimental Hadronic Regge trajectories. Yahiaoui et al. in their cryptographic
work discuss the dynamics and bifurcations of a family of two-dimensional noninvertible
maps. Temnenko in his 4th paper of the series of physics of currents and potentials
addresses Dirac space and vectors.
I want to express my sincere gratitude to the my friend Ignazio Licata for the valu-
able discussions, reviewing and excellent editorial work, and thanks to my friend Hanna
Sabat from the center of theoretical physics and astrophysics for his help in editing the
manuscripts, many thanks to our referees for their valuable comments and notes. We
thank all authors who contributed their articles for this issue.
Ammar Sakaji
Electronic Journal of Theoretical Physics 14, No. 37 (2018) v
EJTP 14, No. 37 (2018) 1–20 Electronic Journal of Theoretical Physics
Projective Limits of State Spaces:Quantum Field Theory Without a Vacuum
Suzanne Lanery∗
Centro de Ciencias Matematicas, Universidad Nacional Autonoma de Mexico, ApartadoPostal 61-3, 58089 Morelia, Michoacan, Mexico
Received 27 December 2017, Accepted 16 March 2018, Published 20 April 2018
Abstract: Instead of formulating the states of a Quantum Field Theory (QFT) as density
matrices over a single large Hilbert space, it has been proposed by Kijowski [20] to construct
them as consistent families of partial density matrices, the latter being defined over small
’building block’ Hilbert spaces. In this picture, each small Hilbert space can be physically
interpreted as extracting from the full theory specific degrees of freedom. This allows to reduce
the quantization of a classical field theory to the quantization of finite-dimensional sub-systems,
thus sidestepping some of the common ambiguities (specifically, the issues revolving around
the choice of a ’vacuum state’), while obtaining robust and well-controlled quantum states
spaces.The present letter provides a self-contained introduction to this formalism, detailing its
motivations as well as its relations to other approaches to QFT (such as conventional Fock-like
Hilbert spaces, path-integral quantization, and the algebraic formulation). At the same time,
it can serve as a reading guide to the series of more in-depth articles [27, 28, 29, 30].c� Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Quantum Field Theory; Vacuum States; Inequivalent Representations; Geometric
Quantization; Projective Limits; Algebras of Observables
PACS (2010): 02.40.Yy; 03.50.-z; 03.70.+k; 04.62.+v; 04.60.Ds
1. Motivation: Quantization Ambiguities in Quantum Field The-
ory
Many choices have to be made in the quantization of a classical theory. Assuming one
is following the canonical quantization path (see section 5. for further discussion of the
relevance for path-integral approaches of the issues discussed here), the first step is to
choose a complete set of basic variables for the theory. Heuristically, these are the vari-
∗ Email: [email protected]
EJTP 14, No. 37 (2018) 1–20 Electronic Journal of Theoretical Physics
Projective Limits of State Spaces:Quantum Field Theory Without a Vacuum
Suzanne Lanery∗
Centro de Ciencias Matematicas, Universidad Nacional Autonoma de Mexico, ApartadoPostal 61-3, 58089 Morelia, Michoacan, Mexico
Received 27 December 2017, Accepted 16 March 2018, Published 20 April 2018
Abstract: Instead of formulating the states of a Quantum Field Theory (QFT) as density
matrices over a single large Hilbert space, it has been proposed by Kijowski [20] to construct
them as consistent families of partial density matrices, the latter being defined over small
’building block’ Hilbert spaces. In this picture, each small Hilbert space can be physically
interpreted as extracting from the full theory specific degrees of freedom. This allows to reduce
the quantization of a classical field theory to the quantization of finite-dimensional sub-systems,
thus sidestepping some of the common ambiguities (specifically, the issues revolving around
the choice of a ’vacuum state’), while obtaining robust and well-controlled quantum states
spaces.The present letter provides a self-contained introduction to this formalism, detailing its
motivations as well as its relations to other approaches to QFT (such as conventional Fock-like
Hilbert spaces, path-integral quantization, and the algebraic formulation). At the same time,
it can serve as a reading guide to the series of more in-depth articles [27, 28, 29, 30].c� Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Quantum Field Theory; Vacuum States; Inequivalent Representations; Geometric
Quantization; Projective Limits; Algebras of Observables
PACS (2010): 02.40.Yy; 03.50.-z; 03.70.+k; 04.62.+v; 04.60.Ds
1. Motivation: Quantization Ambiguities in Quantum Field The-
ory
Many choices have to be made in the quantization of a classical theory. Assuming one
is following the canonical quantization path (see section 5. for further discussion of the
relevance for path-integral approaches of the issues discussed here), the first step is to
choose a complete set of basic variables for the theory. Heuristically, these are the vari-
∗ Email: [email protected]
2 1–20
ables for which the semi-classical limit will work best, hence their choice should ideally
reflect the observables against which the classical theory of interest has been best tested
and confirmed.
The next step is to find a representation of these basic variables as operators on a
suitable Hilbert space H, namely a mapping f �→ f such that
[f , g
]= i {f, g} (1)
(where [ · , · ] denotes the commutator of operators, while { · , · } denotes the Poisson
brackets of classical observables). At this point, quantum field theory (in a broad sense,
namely quantum theories meant to encompass infinitely many degrees of freedom) differs
crucially from quantum mechanics (dealing with the quantum counterparts of classical
systems that have finitely many degrees of freedom). The tools from geometric quanti-
zation [45] (that we will discuss further in subsection 2.2) provide a clear and detailed
understanding of the canonical quantization of finite dimensional systems, including a
parametrization of available choices (aka. quantization ambiguities). In some cases, it
may even turn out that there is no choice at all, because the Poisson-algebra of interest
admits only one suitable representation: this is for example the content of the Stone-von-
Neumann theorem [41, 42, 39] in the case of linear systems.
By contrast, the representation theory for infinite dimensional system tends to be
very involved. Even in the simplest case of a free scalar field on Minkowski spacetime,
it is known that there exist infinitely many inequivalent representations, and although it
has been possible, in this very special case, to fully classify them [15], this classification
is so complex that it gives little insight on how to choose one. As a way out, a pragmatic
way of selecting a good representation among these too numerous options is to single out
a distinguished quantum state, the vacuum: it is indeed possible, via the so-called GNS
construction [11, 38] to ’seed’ a full representationHΩ from a single state Ω (to specify the
latter, even before we are equipped with a Hilbert space, we can give the corresponding
expectation values of all products of the basic variables, aka. the n-point functions, see
[17, part III, def. 2.2.8]). This approach has established itself as the standard way to think
about quantum field theory, at least in the context of Minkowski spacetime, where the
vacuum may be selected by requiring it to be invariant under all spacetime symmetries
(ie. under the Poincare group).
However, one should keep inmind that the initial choice of vacuum is deeply imprinted
in the thus obtained representation. The only quantum states that can be written as (pure
or statistical) states on HΩ are those that barely differ from the vacuum: at most discrete
quantum excitations on top of the state Ω are allowed. The set of all states living on
the representation HΩ is referred to as the vacuum sector, in acknowledgment of the fact
that there are many more quantum states beyond it (falling out of it because they lie
too far away from the chosen vacuum), among whose some may actually be interesting
for specific purposes [17, part V]. An implication of the relative smallness of the vacuum
sector is that the vacuum state need to be closely tailored to the dynamics : otherwise, the
time evolution would immediately kick the states out of HΩ (a precise statement of this
1–20 3
heuristic expectation is given, for Poincare-invariant QFTs, by the Haag no-go theorem,
[16]).
A radical alternative, prompted by the lack of a natural vacuum in the case of quantum
field theory on curved spacetime, is to use as state space the whole set of possible quantum
states over the chosen basic observables (each such state being specified, as explained
above, by the expectations values it prescribes for all products of observables). This
approach can be followed in the context of Algebraic Quantum Field Theory (AQFT,
[17, 18]): by shifting the focus from a particle picture to the local and causal structure
of the quantum theory, AQFT provides tools to discuss the properties of quantum fields
in the absence of an underlying Hilbert space. The aim of the present letter is to argue
that a projective definition of quantum field theory, as was introduced by Jerzy Kijowski
[20] and further developed by Andrzej Oko�low [32, 34, 33], can provide a middle way
between the conventional vacuum-based approach and the full algebraic one, retaining
a constructive description of the quantum state space (subsections 3.2 and 3.3) while
keeping enough flexibility to accommodate a wide class of quantum states (subsection 3.1)
and to decouple the subsequent implementation of the dynamics from the initial building
of the state space (section 4.).
The work summarized in the following sections (and developed in details in [27, 28,
29, 30]) was notably motivated by the specific difficulties encountered when one tries
to formulate background independent quantum field theories, rather than theories on a
(possibly curved) background spacetime (eg. to quantize general relativity itself in a non-
perturbative way [2, 40]). It turns out that for background independent gauge theories
(at least those with compact gauge group), there does exist a preferred vacuum state,
the Ashtekar-Lewandowski vacuum [3, 4], which is uniquely selected precisely by the re-
quirement of background independence [31, 10]. Unfortunately, this vacuum has some
unwanted properties. One of them is that it is an eigenstate of the variable conjugate to
the gauge field, rather than a coherent state like the usual Fock vacuum. Since states in
the vacuum sector cannot differ too much from the vacuum, this makes it difficult to find
semi-classical states among them [12, 22]. Another problem is that the GNS representa-
tion built on this vacuum lives on a non-separable Hilbert space. This particular issue
may or may not go away once we identify quantum states that only differ by a change
of coordinates (depending on how precisely this identification is carried out, see [40, 8])
but in any cases it can lead to technical difficulties [5]. Paradoxically, non-separable
Hilbert spaces seem too small, because their orthonormal basis need uncountably many
basis vectors, making it tempting to consider uncountable linear combinations while only
countable ones are allowed: in other words, it is in this case even more likely that phys-
ically interesting states will lie out of the vacuum sector. In an effort to overcome these
difficulties, the projective quantization techniques that we will review in the next section
have been applied to this kind of theories, resulting in a quantum state space that may
have applications to the study of the semi-classical and cosmological sectors of quantum
gravity [26].
Electronic Journal of Theoretical Physics 14, No. 37 (2018) 1–20
2 1–20
ables for which the semi-classical limit will work best, hence their choice should ideally
reflect the observables against which the classical theory of interest has been best tested
and confirmed.
The next step is to find a representation of these basic variables as operators on a
suitable Hilbert space H, namely a mapping f �→ f such that
[f , g
]= i {f, g} (1)
(where [ · , · ] denotes the commutator of operators, while { · , · } denotes the Poisson
brackets of classical observables). At this point, quantum field theory (in a broad sense,
namely quantum theories meant to encompass infinitely many degrees of freedom) differs
crucially from quantum mechanics (dealing with the quantum counterparts of classical
systems that have finitely many degrees of freedom). The tools from geometric quanti-
zation [45] (that we will discuss further in subsection 2.2) provide a clear and detailed
understanding of the canonical quantization of finite dimensional systems, including a
parametrization of available choices (aka. quantization ambiguities). In some cases, it
may even turn out that there is no choice at all, because the Poisson-algebra of interest
admits only one suitable representation: this is for example the content of the Stone-von-
Neumann theorem [41, 42, 39] in the case of linear systems.
By contrast, the representation theory for infinite dimensional system tends to be
very involved. Even in the simplest case of a free scalar field on Minkowski spacetime,
it is known that there exist infinitely many inequivalent representations, and although it
has been possible, in this very special case, to fully classify them [15], this classification
is so complex that it gives little insight on how to choose one. As a way out, a pragmatic
way of selecting a good representation among these too numerous options is to single out
a distinguished quantum state, the vacuum: it is indeed possible, via the so-called GNS
construction [11, 38] to ’seed’ a full representationHΩ from a single state Ω (to specify the
latter, even before we are equipped with a Hilbert space, we can give the corresponding
expectation values of all products of the basic variables, aka. the n-point functions, see
[17, part III, def. 2.2.8]). This approach has established itself as the standard way to think
about quantum field theory, at least in the context of Minkowski spacetime, where the
vacuum may be selected by requiring it to be invariant under all spacetime symmetries
(ie. under the Poincare group).
However, one should keep inmind that the initial choice of vacuum is deeply imprinted
in the thus obtained representation. The only quantum states that can be written as (pure
or statistical) states on HΩ are those that barely differ from the vacuum: at most discrete
quantum excitations on top of the state Ω are allowed. The set of all states living on
the representation HΩ is referred to as the vacuum sector, in acknowledgment of the fact
that there are many more quantum states beyond it (falling out of it because they lie
too far away from the chosen vacuum), among whose some may actually be interesting
for specific purposes [17, part V]. An implication of the relative smallness of the vacuum
sector is that the vacuum state need to be closely tailored to the dynamics : otherwise, the
time evolution would immediately kick the states out of HΩ (a precise statement of this
1–20 3
heuristic expectation is given, for Poincare-invariant QFTs, by the Haag no-go theorem,
[16]).
A radical alternative, prompted by the lack of a natural vacuum in the case of quantum
field theory on curved spacetime, is to use as state space the whole set of possible quantum
states over the chosen basic observables (each such state being specified, as explained
above, by the expectations values it prescribes for all products of observables). This
approach can be followed in the context of Algebraic Quantum Field Theory (AQFT,
[17, 18]): by shifting the focus from a particle picture to the local and causal structure
of the quantum theory, AQFT provides tools to discuss the properties of quantum fields
in the absence of an underlying Hilbert space. The aim of the present letter is to argue
that a projective definition of quantum field theory, as was introduced by Jerzy Kijowski
[20] and further developed by Andrzej Oko�low [32, 34, 33], can provide a middle way
between the conventional vacuum-based approach and the full algebraic one, retaining
a constructive description of the quantum state space (subsections 3.2 and 3.3) while
keeping enough flexibility to accommodate a wide class of quantum states (subsection 3.1)
and to decouple the subsequent implementation of the dynamics from the initial building
of the state space (section 4.).
The work summarized in the following sections (and developed in details in [27, 28,
29, 30]) was notably motivated by the specific difficulties encountered when one tries
to formulate background independent quantum field theories, rather than theories on a
(possibly curved) background spacetime (eg. to quantize general relativity itself in a non-
perturbative way [2, 40]). It turns out that for background independent gauge theories
(at least those with compact gauge group), there does exist a preferred vacuum state,
the Ashtekar-Lewandowski vacuum [3, 4], which is uniquely selected precisely by the re-
quirement of background independence [31, 10]. Unfortunately, this vacuum has some
unwanted properties. One of them is that it is an eigenstate of the variable conjugate to
the gauge field, rather than a coherent state like the usual Fock vacuum. Since states in
the vacuum sector cannot differ too much from the vacuum, this makes it difficult to find
semi-classical states among them [12, 22]. Another problem is that the GNS representa-
tion built on this vacuum lives on a non-separable Hilbert space. This particular issue
may or may not go away once we identify quantum states that only differ by a change
of coordinates (depending on how precisely this identification is carried out, see [40, 8])
but in any cases it can lead to technical difficulties [5]. Paradoxically, non-separable
Hilbert spaces seem too small, because their orthonormal basis need uncountably many
basis vectors, making it tempting to consider uncountable linear combinations while only
countable ones are allowed: in other words, it is in this case even more likely that phys-
ically interesting states will lie out of the vacuum sector. In an effort to overcome these
difficulties, the projective quantization techniques that we will review in the next section
have been applied to this kind of theories, resulting in a quantum state space that may
have applications to the study of the semi-classical and cosmological sectors of quantum
gravity [26].
Electronic Journal of Theoretical Physics 14, No. 37 (2018) 1–20
4 1–20
Mη′′
Mη′
Mη
πη′′→η′
πη′′→η
πη′→η
Mη′′
Mη′′→η′ × Mη′
Mη′′→η′ ×Mη′→η × Mη Mη′′→η× Mη
≈
Fig. 1 Three-spaces consistency for projective systems (left side), reformulated in terms offactorizations (right side)
2. Systematic Quantization of Infinite-dimensional Systems
2.1 Building an Infinite-dimensional Theory from a Collection of Partial
Descriptions
The key observation underlying Kijowski’s projective formalism [20, 33] is that a given
experiment can only measure finitely many observables. Thus, we never need to consider
at once the full, infinite-dimensional phase space M∞ of a field theory: it is sufficient
to work in a small, partial phase space Mη that extracts from M∞ just the degrees of
freedom (dof.2) relevant for the experiment at hand (throughout the present letter, the
symbol η will be used to denote a selection of finitely many dof. out of the full theory,
and we will call η a label).
In order to use such a collection of finite-dimensional partial phase spaces(Mη
)ηto
completely specify a field theory, we need to ensure that the different partial theories are
consistent with each other [27, subsection 2.1]:
1. first, we need a way to express the relations between the dof. in different labels. We
will write η � η� if all dof. contained in η are also contained in η� (we will also say
that η is coarser as η�, or that η� is finer as η). This means that any observable fηon Mη corresponds to an observable fη′ on Mη′ , and, by duality3, that there exists a
projection πη′→η from Mη′ to Mη such that
fη′ = fη ◦ πη′→η
2. the predictions for a given experiment, as calculated in a partial theory η, should be
independent of the choice of η (provided η is fine enough to hold all relevant dof.).
Thus, in particular, the Poisson brackets between two observables fη and gη on Mη
2 By a dof. we mean a pair of conjugate variables.3 To see that πη′→η is uniquely specified once we know the mapping fη �→ fη′ between observables, one
can consider a complete set of observables (aka. coordinates) on Mη.
1–20 5
should agree with the Poisson brackets between the corresponding observables on a
finer Mη′ . Expressed in terms of the just introduced projection πη′→η, this reads
{fη ◦ πη′→η, gη ◦ πη′→η} = {fη, gη} ◦ πη′→η (2)
3. it should be possible to consider composite experiments made of two (or more) sub-
experiments, each of which can be described within a different partial theory4. In
other words, for any η, η�, there should exist η�� such that η, η� � η��. This property is
called directedness of the set of labels.
4. the relation between any two partial theories should be unambiguous. Thanks to the
just mentioned directedness property, this can be ensured simply by requiring that the
projections defined among three increasingly refined partial theories match as shown
on the left part of fig. 1.
In mathematical terms, this list of requirements can be summarized by saying that the
collection(Mη
)ηforms a projective (aka. inverse) system and it ensures that M∞ can
be reconstructed from(Mη
)η(more precisely, a space Mlim can be constructed as the
so-called projective limit of this system, that will, in general, be a distributional extension
of M∞, see [27, def. 2.6 and prop. 2.7]).
A key point of the construction is that each label η corresponds to a selection of con-
jugate position and momentum variables. This ensures that the projection map πη′→η
between the phase spacesMη′ andMη is unambiguous: as stressed above, it is completely
determined by matching the physical interpretation of the observables in η vs. η�. By
contrast, if the partial theories were labeled by selections of configuration variables only
(as is usual eg. when studying coarse graining in the path-integral formalism), we would
only be provided with projections between the configuration spaces (since phase spaces
can be thought of as cotangent bundles, and forms are naturally pull-backed, rather than
push-forwarded, there is no canonical prescription to lift a projection between configu-
ration spaces to a projection between the associated phase spaces). Furthermore, our
labels do not have to be assembled from pre-assigned pairs of mutually independent,
canonically conjugate variables (aka. modes). In particular, different labels can consist of
the same selection of configuration variables, but paired with different conjugate impul-
sions5: the corresponding partial theories are then interpreted as extracting from the full,
continuum field theory information about different (partially overlapping) dof. The di-
rectedness requirement (which is much weaker than the requirement of a preferred mode
decomposition, and is indeed fulfilled in a range of situations where the latter would not
4 We are discussing the classical theory here. The quantum theory is more subtle, since one could
argue that, due to the principle of complementarity, some sub-experiments may be mutually excluding.
However, the case for the directedness of the label set can still be made, see [28, section 1].5 This is manifest eg. in the treatment of gauge theories in [26]: selections of configuration variables are
represented by graphs (generalized lattices) and these need to be decorated with dual surfaces to fully
specify the impulsions (aka. electric fluxes) captured by a given partial theory (see the discussion at the
beginning of [26, section II]).
Electronic Journal of Theoretical Physics 14, No. 37 (2018) 1–20