EJTP ElEctronic Journal of thEorEtical Physics

Volume 14 April, 2018

Number 37

http://www.ejtp.com E-mail:info@ejtp.com

editorsIgnazio LicataAmmar Sakaji

Aracne editrice

www.aracneeditrice.itinfo@aracneeditrice.it

Copyright © MMXVIIGioacchino Onorati editore S.r.l. – unipersonale

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via Vittorio Veneto, 2000020 Canterano (RM)

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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: submission.ejtp@gmail.com

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: submission.ejtp@gmail.com

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: vitiello@sa.infn.it

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: vitiello@sa.infn.it

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: slanery@matmor.unam.mx

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: slanery@matmor.unam.mx

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