پژوهش های نوین فیزیک

هامیلتونی و براکت پوآسون نسبیتی

نویسنده

ابوالفضل جعفری

دانشگاه شهرکرد

https://doi.org/10.66224/jmrph.10.1.51
چکیده
در رژیم نسبیتی و برای زمانی‌ که لاگرانژی در دسترس است، روشی برای استخراج نسخه نسبیتی هامیلتونی پیشنهاد می‌دهیم و نشان می‌دهیم که یک تبدیل کوچک در تعریف لاگرانژی برای رسیدن به این هدف کافی خواهد بود. توانایی هامیلتونی به دست آمده در توصیف نظریه میدان‌ها را بررسی نموده و با استناد به تکانه‌های کانونیک و مکانیکی، نظریه‌های مورد نظر را بازتولید خواهیم کرد. از هامیلتونی نسبیتی به دست آمده در دو رهیافت متفاوت نظریه میدان‌های بوزونی و فرمیونی را می‌سازیم و هم‌ریشه بودن آن‌ها را روشن می‌کنیم. مکانیزم تبدیل لژاندرِ روی لاگرانژی از فضای پیکربندی به فضای فاز و بالعکس را به دست آورده و ضرب پوآسون را تعمیم می‌دهیم. بعد از استخراج هامیلتونی و معادلات حرکت منسوب به هامیلتون نسبیتی، فضای فاز را گسترش داده و نشان می‌دهیم که ایجاد فضای فاز دوگان به همراه باز تعریف مشتق‌گیرها روی فضای فاز دوگان و فضای فاز اصلی و ایجاد تغییر در ساختار ضرب پوآسون لازم خواهد بود تا به ضرب پوآسون نسبیتی نزدیک شویم. مشخص می‌شود که وجود فضای فازِ کامل ناوردای لورنتسی، مستلزم اصلاح نظریه کلاسیک و رسیدن به هامیلتونی نسبیتی نیز موثر است.

کلیدواژه‌ها

سیستم‌های نسبیتی
لاگرانژی و هامیلتونی نسبیتی
نظریه میدان‌های بوزونی و فرمیونی
تبدیل لژاندر
فضای دوگان
براکت پوآسون نسبیتی

عنوان مقاله English

Relativistic Hamiltonian and Poisson Bracket

نویسنده English

Abolfazl Jafari
چکیده English

In the relativistic regime, we propose a method for deriving the relativistic version of the Hamiltonian, assuming the Lagrangian is available. We demonstrate that a minor modification in the definition of the Lagrangian is sufficient to achieve this goal. We will investigate the ability of the obtained Hamiltonian to describe field theory, and, by referring to canonical and mechanical momenta, we will reproduce the theories of interest. From the relativistic Hamiltonian obtained through two different approaches, we will construct the theories of bosonic and fermionic fields and clarify their interconnection. We explain the mechanism of the Legendre transformation between configuration space and phase space, and we extend the Poisson product. After deriving the Hamiltonian and the equations of motion attributed to the relativistic Hamiltonian, we extend the phase space and show that the creation of a dual phase space, along with the redefinition of the derivatives on the dual phase space and the original phase space, and the modification of the Poisson product structure will be necessary to approach the relativistic Poisson product. To establish a complete Lorentzian phase space, modifications to classical theory and an effective realization of the relativistic Hamiltonian are necessary.

کلیدواژه‌ها English

Relativistic Systems
Relativistic Lagrangian and Hamiltonian
Bosonic and Fermionic Field Theory
Legendre Transform
Phase Dual Space
Relativistic Poisson Bracket
[1] Yoel Tikochinsky, International Journal of Theoretical Physics, VoL 33, No. 8, 1994 [DOI:10.1007/BF00672685]
[1] Yoel Tikochinsky, International Journal of Theoretical Physics, VoL 33, No. 8, 1994 [DOI:10.1007/BF00672685]
[2] Paul Bracken, International Journal of Theoretical Physics, Vol. 37, No. 5, 1998 [DOI:10.1023/A:1026680205982]
[2] Paul Bracken, International Journal of Theoretical Physics, Vol. 37, No. 5, 1998 [DOI:10.1023/A:1026680205982]
[3] H. Goldstein, Charles P. Poole and John L. Safko "Classical Mechanics", published by Pearson Education, Inc., publishing as Addison-Wesley, Copyright, 2002
[3] H. Goldstein, Charles P. Poole and John L. Safko "Classical Mechanics", published by Pearson Education, Inc., publishing as Addison-Wesley, Copyright, 2002
[4] V. I. Arnol'd, K. Vogtmann, A. Weinstein, "Mathematical Methods of Classical Mechanics", Springer-Verlag New York Inc. 1989
[4] V. I. Arnol'd, K. Vogtmann, A. Weinstein, "Mathematical Methods of Classical Mechanics", Springer-Verlag New York Inc. 1989
[5] William L. Bruke, "Applied Differential Geometry", Cambridge University Press, Cambridge, New York 1985
[5] William L. Bruke, "Applied Differential Geometry", Cambridge University Press, Cambridge, New York 1985
[6] A. Stern and I. Yakushin, "Deformed Wong particles", Phys. Rev. D, Vol 48, N. 10 1993 [DOI:10.1103/PhysRevD.48.4974] [PMID]
[6] A. Stern and I. Yakushin, "Deformed Wong particles", Phys. Rev. D, Vol 48, N. 10 1993 [DOI:10.1103/PhysRevD.48.4974] [PMID]
[7] Freeman J. Dyson, "Feynman's Proof of the Maxwell's Equations", Am. J. Phys. 58 3 209 1990 [DOI:10.1119/1.16188]
[7] Freeman J. Dyson, "Feynman's Proof of the Maxwell's Equations", Am. J. Phys. 58 3 209 1990 [DOI:10.1119/1.16188]
[8] J. J. Sakurai, "Advanced Quantum Mechanics", Addison-Wesley Publishing Company, Inc. 1994
[8] J. J. Sakurai, "Advanced Quantum Mechanics", Addison-Wesley Publishing Company, Inc. 1994
[9] L. E. Parker and David J. Toms, "Quantum Field Theory in Curved Spacetime", United States of America by Camberidge University Press, New York, 2009 [DOI:10.1017/CBO9780511813924]
[9] L. E. Parker and David J. Toms, "Quantum Field Theory in Curved Spacetime", United States of America by Camberidge University Press, New York, 2009 [DOI:10.1017/CBO9780511813924]
[10] N. D. Birrell and P. C. W. Davies, "Quantum Fields On Curved Space", Cambridge University Press 1982 [DOI:10.1017/CBO9780511622632]
[10] N. D. Birrell and P. C. W. Davies, "Quantum Fields On Curved Space", Cambridge University Press 1982 [DOI:10.1017/CBO9780511622632]
[11] Stephen A. Fulling, "Aspects of Quantum Field Theory in Curved Space-Time", Cambridge University Press 1989 [DOI:10.1017/CBO9781139172073]
[11] Stephen A. Fulling, "Aspects of Quantum Field Theory in Curved Space-Time", Cambridge University Press 1989 [DOI:10.1017/CBO9781139172073]
[12] Bryce S. DeWitt, "Quantum Field Theory in Curved Spacetime", North Holland Publishing Company, Amesterdam 1975 [DOI:10.1063/1.2947455]
[12] Bryce S. DeWitt, "Quantum Field Theory in Curved Spacetime", North Holland Publishing Company, Amesterdam 1975 [DOI:10.1063/1.2947455]
[13] Claude Itzykson and Jean Bernard Zuber, "Quantum Field Theory", Dover Publications, INC. Mineola, New York, 1980
[13] Claude Itzykson and Jean Bernard Zuber, "Quantum Field Theory", Dover Publications, INC. Mineola, New York, 1980
[14] N. N. Bogoliubov and D. V. Shirkov, "Quantum Fields", Benjamin/Cummings Publishing Company, Inc. 1983.
[14] N. N. Bogoliubov and D. V. Shirkov, "Quantum Fields", Benjamin/Cummings Publishing Company, Inc. 1983.
[15] Jean-Paul Dufour, Nguyen Tien Zung, "Poisson Structures and Their Normal Forms", Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland 2005
[15] Jean-Paul Dufour, Nguyen Tien Zung, "Poisson Structures and Their Normal Forms", Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland 2005
[16] J. E. Marsden, R. Montgomery, P. J. Morrison and and W. B. Thompson, "Covariant Poisson Brackets for Classical Fields", Annals of Physics 169, 2947 1986 [DOI:10.1016/0003-4916(86)90157-0]
[16] J. E. Marsden, R. Montgomery, P. J. Morrison and and W. B. Thompson, "Covariant Poisson Brackets for Classical Fields", Annals of Physics 169, 2947 1986 [DOI:10.1016/0003-4916(86)90157-0]
[17] J. Kowalski Glikman, "Approaches to Quantum Gravit: Toward a New Understanding of Space, Time and Matter", ed. Daniele Oriti. Cambridge University Press 2009
[17] J. Kowalski Glikman, "Approaches to Quantum Gravit: Toward a New Understanding of Space, Time and Matter", ed. Daniele Oriti. Cambridge University Press 2009
[18] . Bird, D. J. et al., "Detection of a Cosmic Ray with Measured Energy Well beyond the Expected Spectral Cutoff due to Cosmic Microwave Radiation", Astrophys. J. 441, 144-150 1995 [DOI:10.1086/175344]
[18] . Bird, D. J. et al., "Detection of a Cosmic Ray with Measured Energy Well beyond the Expected Spectral Cutoff due to Cosmic Microwave Radiation", Astrophys. J. 441, 144-150 1995 [DOI:10.1086/175344]
[19] Magueijo, J. & Smolin, L. "Lorentz Invariance with an Invariant Energy Scale", Phys. Rev. Lett. 88, 190403 2002 [DOI:10.1103/PhysRevLett.88.190403] [PMID]
[19] Magueijo, J. & Smolin, L. "Lorentz Invariance with an Invariant Energy Scale", Phys. Rev. Lett. 88, 190403 2002 [DOI:10.1103/PhysRevLett.88.190403] [PMID]
[20] Kowalski-Glikman, J. & Nowak, S. "Doubly Special Relativity theories as different bases of κ-Poincar'e algebra", Phys.Lett. B 539 126-132 2002 [DOI:10.1016/S0370-2693(02)02063-4]
[20] Kowalski-Glikman, J. & Nowak, S. "Doubly Special Relativity theories as different bases of κ-Poincar'e algebra", Phys.Lett. B 539 126-132 2002 [DOI:10.1016/S0370-2693(02)02063-4]
[21] Nejad, S.A., Dehghani, M. & Monemzadeh, M. "Lagrange multiplier and Wess-Zumino variable as extra dimensions in the torus universe", J. High Energ. Phys 17 2018 [DOI:10.1007/JHEP01(2018)017]
[21] Nejad, S.A., Dehghani, M. & Monemzadeh, M. "Lagrange multiplier and Wess-Zumino variable as extra dimensions in the torus universe", J. High Energ. Phys 17 2018 [DOI:10.1007/JHEP01(2018)017]
[22] M. Dehghani, S. Abarghouei Nejad and M. Monemzadeh, "Modified Anyonic Particle and Its Fundamental Gauge Symmetries", Hindawi Advances in High Energy Physics Vol. 2020, 2020 [DOI:10.1155/2020/7126374]
[22] M. Dehghani, S. Abarghouei Nejad and M. Monemzadeh, "Modified Anyonic Particle and Its Fundamental Gauge Symmetries", Hindawi Advances in High Energy Physics Vol. 2020, 2020 [DOI:10.1155/2020/7126374]

صفحه اصلی

راهنمای نویسندگان

ارسال مقاله

داوران

تماس با ما