熱場の量子論のソースを表示

[[理論物理学]]では、'''熱場の量子論'''(thermal quantum field theory)、あるいは、'''有限温度の場の理論'''(finite temperature field theory)は、有限[[温度]]での[[量子場理論]]の物理的観測量の期待値を計算する一連の方法である。
<!--In [[theoretical physics]], '''thermal quantum field theory'''  ('''thermal field theory''' for short) or '''finite temperature field theory''' is a set of methods to calculate expectation values of physical observables of a [[quantum field theory]] at finite [[temperature]].-->

{{仮リンク|松原の定式化|en|Matsubara formalism}}(Matsubara formalism)では、（[[フェリックス・ブロッホ]](Felix Bloch)によれば<ref name="Bloch1932">{{Cite journal
|author=Bloch, F.
|title=Zur Theorie des Austauschproblems und der Remanenzerscheinung der Ferromagnetika
|journal=Z. Phys.
|volume=74
|issue=5-6
|pages=295–335
|year=1932
|doi=10.1007/BF01337791
|bibcode = 1932ZPhy...74..295B }}</ref>）基本的考え方は、熱アンサンブルの中の作用素の期待値は、
:<math> \langle A\rangle=\frac{\mbox{Tr}\, [\exp(-\beta H) A]}{\mbox{Tr}\, [\exp(-\beta H)]}</math>
により通常の[[量子場理論]]の[[期待値]]として記述することができる<ref>{{Cite book|author=Jean Zinn-Justin|title=Quantum Field Theory and Critical Phenomena|publisher=Oxford University Press|year=2002|isbn=978-0-19-850923-3}}</ref>。ここに、構成は、[[虚時間]] <math>\tau=-it(0\leq\tau\leq\beta)</math> により発展する。従って、[[ユークリッド距離|ユークリッド計量]]を持つ[[時空]]へ切り替えることができる。ユークリッド計量を持つ時空の上では、（[[自然単位系]]を <math>\hbar =  1</math> を前提として）周期 <math>\beta = 1/(kT)</math> を持つユークリッド時間発展方向に関して、トレース (Tr) が[[ボゾン|ボゾニック]]な場は周期的であり、[[フェルミオン|フェルミオニック]]な場は反周期的であることを要求する。そこでは、コンパクトなユークリッド時間を持つ[[経路積分]]や[[ファインマン図]]のような通常の量子場の理論として、同じツールを使い計算を実行することができる。正規順序の定義を変更する必要があることに注意する<ref name="Evans1996">{{Cite journal
|author=T.S. Evans and D.A. Steer, 
|title= Wick's theorem at finite temperat
|journal=Nucl.Phys.B
|volume=474
|issue=2
|pages=481–496
|year=1996
|doi=10.1016/0550-3213(96)00286-6
|arxiv = hep-ph/9601268 |bibcode = 1996NuPhB.474..481E }}</ref>。[[波数空間|運動量空間]]では、このために、連続周期が離散的な虚周期（松原周期） <math>v_n =  n / \beta </math> へ置き換り、[[ド・ブロイ波|ド・ブロイ関係式]]を通して、離散化された熱エネルギースペクトル <math>E_n =  n K T </math> へ置き換わる。このことが、有限温度での量子場の振る舞いの研究に有益なツールであることを示している<ref>D.A. Kirznits JETP Lett. 15 (1972) 529.</ref><ref>D.A. Kirznits and A.D. Linde, Phys. Lett. B42 (1972) 471; it Ann. Phys.
 101 (1976) 195.</ref><ref name="Weinberg1974">{{Cite journal
|author=Weinberg, S. 
|title= Gauge and Global Symmetries at High Temperature
|journal=Phys. Rev. D
|volume=9
|issue=12
|pages=3357–3378
|year=1974
|doi=10.1103/PhysRevD.9.3357 
|publisher=American Physical Society
|bibcode = 1974PhRvD...9.3357W }}</ref>
<ref name="Dolan1974">{{Cite journal
|author=L. Dolan, and  R. Jackiw
|title=Symmetry behavior at finite temperature
|journal=Phys. Rev. D
|volume=9
|issue=12
|pages=3320–3341
|year=1974
|doi=10.1103/PhysRevD.9.3320
|publisher=American Physical Society
|bibcode = 1974PhRvD...9.3320D }}</ref>。

この考え方は、ゲージ不変性を持つ理論へと一般化されていて、[[ヤン・ミルズ理論]]の非閉じ込め相転移を解明するための中心的なツールとなっている<ref>C. W. Bernard, Phys. Rev. D9 (1974) 3312.</ref><ref>D.J. Gross, R.D. Pisarski and L.G. Yaffe, Rev. Mod. Phys. 53 (1981) 43.</ref>。このユークリッド的な場の理論では、実時間の観測量を、[[解析接続]]することにより、回復することが可能である<ref name="Evans1992">{{Cite journal
|author=T.S. Evans
|title= N-Point Finite Temperature Expectation Values at Real Times
|journal=Nucl.Phys.B
|volume=374
|issue=2
|pages=340–370
|year=1992
|doi=10.1016/0550-3213(92)90357-H
|arxiv=hep-ph/9601268
|bibcode = 1992NuPhB.374..340E }}</ref>。
<!--In the [[Matsubara formalism]],
the basic idea (due to [[Felix Bloch]]<ref name="Bloch1932">{{Cite journal
|author=Bloch, F.
|title=Zur Theorie des Austauschproblems und der Remanenzerscheinung der Ferromagnetika
|journal=Z. Phys.
|volume=74
|issue=5-6
|pages=295–335
|year=1932
|doi=10.1007/BF01337791
|bibcode = 1932ZPhy...74..295B }}</ref>) is that the expectation values of operators in a thermal ensemble
:<math> \langle A\rangle=\frac{\mbox{Tr}\, [\exp(-\beta H) A]}{\mbox{Tr}\, [\exp(-\beta H)]}</math>
may be written as [[expectation value]]s in ordinary [[quantum field theory]]<ref>{{Cite book|author=Jean Zinn-Justin|title=Quantum Field Theory and Critical Phenomena|publisher=Oxford University Press|year=2002|isbn=978-0-19-850923-3}}</ref> where the configuration is evolved by an [[imaginary time]] <math>\tau=-it(0\leq\tau\leq\beta)</math>. One can therefore switch to
a [[spacetime]] with [[Euclidean signature]], where the
above trace (Tr) leads to the requirement that all [[boson]]ic and [[fermion]]ic
fields be periodic and antiperiodic, respectively, with respect to
the Euclidean time direction with periodicity <math>\beta = 1/(kT)</math> (we are assuming  [[natural units]] <math>\hbar =  1</math>). This allows
one to perform calculations with the same tools as in ordinary quantum field theory,
such as [[functional integral]]s and [[Feynman diagram]]s, but with compact Euclidean time.  Note that the definition of normal ordering has to be altered.<ref name="Evans1996">{{Cite journal
|author=T.S. Evans and D.A. Steer, 
|title= Wick's theorem at finite temperature
|journal=Nucl.Phys.B
|volume=474
|issue=2
|pages=481–496
|year=1996
|doi=10.1016/0550-3213(96)00286-6
|arxiv = hep-ph/9601268 |bibcode = 1996NuPhB.474..481E }}</ref>
In [[momentum space]], this leads to the replacement of continuous frequencies by
discrete imaginary (Matsubara) frequencies <math>v_n =  n / \beta </math> and, through the [[de Broglie relation]], to a discretized thermal energy spectrum <math>E_n =  n K T </math>.  This has been shown to be a useful tool
in studying the behavior of quantum field theories at finite temperature. 
<ref>D.A. Kirznits JETP Lett. 15 (1972) 529.</ref>
<ref>D.A. Kirznits and A.D. Linde, Phys. Lett. B42 (1972) 471; it Ann. Phys.
 101 (1976) 195.</ref><ref name="Weinberg1974">{{Cite journal
|author=Weinberg, S. 
|title= Gauge and Global Symmetries at High Temperature
|journal=Phys. Rev. D
|volume=9
|issue=12
|pages=3357–3378
|year=1974
|doi=10.1103/PhysRevD.9.3357 
|publisher=American Physical Society
|bibcode = 1974PhRvD...9.3357W }}</ref>
<ref name="Dolan1974">{{Cite journal
|author=L. Dolan, and  R. Jackiw
|title=Symmetry behavior at finite temperature
|journal=Phys. Rev. D
|volume=9
|issue=12
|pages=3320–3341
|year=1974
|doi=10.1103/PhysRevD.9.3320
|publisher=American Physical Society
|bibcode = 1974PhRvD...9.3320D }}</ref>
It has been generalized to theories with gauge invariance and was a central tool
in the study of a conjectured deconfining phase transition of Yang-Mills theory.
<ref>C. W. Bernard, Phys. Rev. D9 (1974) 3312.</ref>
<ref>D.J. Gross, R.D. Pisarski and L.G. Yaffe, Rev. Mod. Phys. 53 (1981) 43.</ref>
In this Euclidean field theory, real-time observables can be retrieved
by [[analytic continuation]].<ref name="Evans1992">{{Cite journal
|author=T.S. Evans
|title= N-Point Finite Temperature Expectation Values at Real Times
|journal=Nucl.Phys.B
|volume=374
|issue=2
|pages=340–370
|year=1992
|doi=10.1016/0550-3213(92)90357-H
|arxiv=hep-ph/9601268
|bibcode = 1992NuPhB.374..340E }}</ref>-->

架空の虚時間を使うもうひとつの使い方は、2-形式に表すことができる実時間定式化を使うことである<ref name="Landsman1987">{{Cite journal
|author=N.P. Landsman and Ch.G. van Weert  
|title= Real- and imaginary-time field theory at finite temperature and density
|journal=Physics Reports
|volume=145
|issue=3-4
|pages=141–249
|year=1987
|doi=10.1016/0370-1573(87)90121-9
|bibcode = 1987PhR...145..141L }}</ref>。実時間の定式化への経路順序のアプローチは、[[ケルディッシュ形式]](Schwinger-Keldysh formalism)であり、より現代的な変形である<ref name=niemisemenoff>{{Cite journal|author=A.J. Niemi, G.W. Semenoff|title=Finite Temperature Quantum Field Theory in Minkowski Space|journal=Annals Phys|volume=152|pages=105|year=1984|doi=10.1016/0003-4916(84)90082-4|bibcode = 1984AnPhy.152..105N }}</ref>。後者は、大きな負の実時間 <math>t_f</math> へ行き、<math>t_i - i\beta</math> へ戻ることにより、初期実時間（大きな負の値） <math>t_i</math> を <math>t_i - i\beta</math> へ変換して、まっすぐな時間積分路に置き換えることを意味する<ref>{{Cite arxiv
|author=Zinn-Justin, Jean
|title=Quantum field theory at finite temperature: An introduction
|year=2000
|eprint=hep-ph/0005272
}}</ref>。実際、実時間軸にそって、終点 <math>t_i - i\beta</math> へ向かう経路として積分路をとることの重要性が小さくなることが、必要なことのすべてである<ref name="Evans1993">{{Cite journal
|author=T.S. Evans, 
|title= New Time Contour for Equilibrium Real-Time Thermal Field-Theories
|journal=Phys.Rev.D
|volume=47
|issue=10
|pages=R4196-R4198
|year=1993
|doi=10.1103/PhysRevD.47.R4196
|arxiv = hep-ph/9310339 |bibcode = 1993PhRvD..47.4196E }}</ref>。結果としての複素積分路の区分合成は、場が二重化され、より込み入ったファイマン規則となるが、虚時間の定式化の[[解析接続]]の必要性はなくなる。実時間のアプローチのもうひとつは、'''熱場の力学'''として知られる、[[ボゴリューボフ変換]]を用いた作用素を基礎とするアプローチである<ref name="Landsman1987" /><ref>{{Cite journal
|author1=H. Chiu
|author2=H. Umezawa
|authorlink2=Hiroomi Umezawa
|title= A unified formalism of thermal quantum field theory
|journal=International Journal of Modern Physics A
|volume=9
|issue=14
|pages=2363 ff.
|year=1993
|doi=10.1142/S0217751X94000960
|bibcode =   1994IJMPA...9.2363C}}
</ref>。ファインマン図や摂動論と同様に、分散関係式や{{仮リンク|クツォスキー規則|en|Cutkosky rules}}(Cutkosky rules)の有限温度での類似のような他のテクニックも、実時間定式化の中で使われる<ref name=kobessemenoff>{{Cite journal|author=R.L. Kobes, G.W. Semenoff|title=Discontinuities of Green Functions in Field Theory at Finite Temperature and Density|journal=Nucl.Phys.|volume=B260|issue=3-4|pages=714–746|year=1985|doi=10.1016/0550-3213(85)90056-2|bibcode = 1985NuPhB.260..714K }}</ref><ref name=kobessemenoff2>{{Cite journal|author=R.L. Kobes, G.W. Semenoff|title=Discontinuities of Green Functions in Field Theory at Finite Temperature and Density |journal=Nucl.Phys.|volume=B272|issue=2|pages=329–364|year=1986|doi=10.1016/0550-3213(86)90006-4|bibcode = 1986NuPhB.272..329K }}</ref>。

数理物理学で興味の持たれているもうひとつのアプローチは、[[KMS状態]]とともに機能させることである。
<!--The alternative to the use of fictitious imaginary times is to use a real-time formalism which come in two forms.<ref name="Landsman1987">{{Cite journal
|author=N.P. Landsman and Ch.G. van Weert  
|title= Real- and imaginary-time field theory at finite temperature and density
|journal=Physics Reports
|volume=145
|issue=3-4
|pages=141–249
|year=1987
|doi=10.1016/0370-1573(87)90121-9
|bibcode = 1987PhR...145..141L }}</ref>  A path-ordered approach to real-time formalisms includes the [[Keldysh formalism|Schwinger-Keldysh formalism]] and more modern variants.<ref name=niemisemenoff>{{Cite journal|author=A.J. Niemi, G.W. Semenoff|title=Finite Temperature Quantum Field Theory in Minkowski Space|journal=Annals Phys|volume=152|pages=105|year=1984|doi=10.1016/0003-4916(84)90082-4|bibcode = 1984AnPhy.152..105N }}</ref>
The latter involves replacing a straight time contour from (large negative) real
initial time <math>t_i</math> to <math>t_i - i\beta</math> by one that first runs to (large positive) real time <math>t_f</math> and then suitably back to <math>t_i - i\beta</math>.<ref>{{Cite arxiv
|author=Zinn-Justin, Jean
|title=Quantum field theory at finite temperature: An introduction
|year=2000
|eprint=hep-ph/0005272
|class=hep-ph
}}</ref> In fact all that is needed is one section running along the real time axis as the route to 
the end point, <math>t_i - i\beta</math>, is less important.<ref name="Evans1993">{{Cite journal
|author=T.S. Evans, 
|title= New Time Contour for Equilibrium Real-Time Thermal Field-Theories
|journal=Phys.Rev.D
|volume=47
|issue=10
|pages=R4196-R4198
|year=1993
|doi=10.1103/PhysRevD.47.R4196
|arxiv = hep-ph/9310339 |bibcode = 1993PhRvD..47.4196E }}</ref>
The piecewise composition
of the resulting complex time contour leads to a doubling of fields and more complicated
Feynman rules, but obviates the need of [[analytic continuation]]s of the imaginary-time formalism. The alternative approach to real-time formalisms is an operator based approach using [[Bogoliubov transformation]]s, known as '''thermo field dynamics'''.<ref name="Landsman1987" /><ref>{{Cite journal
|author1=H. Chiu
|author2=H. Umezawa
|authorlink2=Hiroomi Umezawa
|title= A unified formalism of thermal quantum field theory
|journal=International Journal of Modern Physics A
|volume=9
|issue=14
|pages=2363 ff.
|year=1993
|doi=10.1142/S0217751X94000960
|bibcode =   1994IJMPA...9.2363C}}
</ref>
As well as Feynman diagrams and perturbation theory, other techniques such as dispersion relations
and the finite temperature analog of [[Cutkosky rules]] can also be used in the real time formulation <ref name=kobessemenoff>{{Cite journal|author=R.L. Kobes, G.W. Semenoff|title=Discontinuities of Green Functions in Field Theory at Finite Temperature and Density|journal=Nucl.Phys.|volume=B260|issue=3-4|pages=714–746|year=1985|doi=10.1016/0550-3213(85)90056-2|bibcode = 1985NuPhB.260..714K }}</ref>
.<ref name=kobessemenoff2>{{Cite journal|author=R.L. Kobes, G.W. Semenoff|title=Discontinuities of Green Functions in Field Theory at Finite Temperature and Density |journal=Nucl.Phys.|volume=B272|issue=2|pages=329–364|year=1986|doi=10.1016/0550-3213(86)90006-4|bibcode = 1986NuPhB.272..329K }}</ref>

An alternative approach which is of interest to mathematical physics is to work with
[[KMS state]]s.-->

== 参照項目 ==
*{{仮リンク|松原周波数|en|Matsubara frequency}}(Matsubara frequency)

==参考文献==
<ref name=fetterwalecka>{{Cite book|author=Alexander L. Fetter, John Dirk Walecka|title=Quantum Theory of Many-Particle Systems|publisher=Dover Publications|year=2003|isbn=978-0-486-42827-7}}</ref>

<references/>

{{DEFAULTSORT:ねつはのりようしろん}}
[[Category:場の量子論]]
[[Category:統計力学]]