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Time-Ordered Exponential for Unbounded Operators with Applications to Quantum Field Theory

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Time-Ordered Exponential for Unbounded Operators with Applications to Quantum Field Theory Title Time-ordered Exponential for Unbounded Operators with Applications to Quantum Field Theory Author(s) 二口, 伸一郎 Citation 北海道大学. 博士(理学) 甲第11798号 Issue Date 2015-03-25 DOI 10.14943/doctoral.k11798 Doc URL http://hdl.handle.net/2115/58756 Type theses (doctoral) File Information Shinichiro_Futakuchi.pdf Instructions for use Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP Time-ordered Exponential for Unbounded Operators with Applications to Quantum Field Theory (非有界作用素に対する time-ordered exponential と 場の量子論への応用) A dissertation submitted to Hokkaido University for the degree of Doctor of Sciences presented by Shinichiro Futakuchi Department of Mathematics Graduate School of Science Hokkaido University advised by Asao Arai March 2015 Abstract Time-ordered exponential is a fundamental tool in theoretical physics and mathematical physics, and often used in quantum theory to give the perturbative expansion of significant objects, such as the time evolution, the n-point correlation functions, and the scattering amplitudes. The time-ordered exponential generated by a bounded operator-valued function has already been well researched, but the one generated by an unbounded operator-valued function has not fully investigated so far; the mathematical theory which is applicable to the analysis of concrete models of quantum field theory has been inadequately studied. The first main purpose of this work is to provide a general mathematical theory on time-ordered exponential. The second main purpose is to construct concrete quantum field models and to analyze them rigorously. In this paper, we study the following: (I) construction of dynamics for non- symmetric Hamiltonians, (II) Gupta-Bleuler formalism, (III) Gell-Mann{Low formula, (IV) criteria for essential self-adjointness. (I) In general, to quantize canonically a gauge theory in a Lorentz-covariant gauge such as the Lorenz gauge, it is necessary to adopt a vector space with an indefinite metric as a vector space of quantum mechanical state vectors, in order to realize the canonical commutation relations. The most important examples that cause this situation contain the mathematical model of quantum electrodynamics (QED) quantized in the Lorenz gauge. In this paper, we adopt a method called η-formalism that defines an indefinite metric through a Hilbert space with an inner product. Then the Hamiltonian is self-adjoint with respect to the indefinite metric but not symmetric and not even normal with respect to the inner product. By using the time-ordered exponential, we prove under certain assumptions that there exists a solution of the Schr¨odingeror the Heisenberg equation of motion generated by a linear operator acting in a Hilbert space, which may be unbounded, not symmetric, or not normal. (II) As mentioned above, to quantize QED in the Lorenz gauge, the metric is indefinite. Then the Gupta-Bleuler formalism is used to restrict the total state space to the vector space of physical states which is a non-negative inner product space. Using the results obtained in (I), we apply the Gupta-Bleuler formalism to the Dirac-Maxwell model quantized in the Lorenz gauge, which describes a quantum system of Dirac particles and a gauge field minimally interacting each other. (III) The Gell-Mann{Low formula plays an essential role in quantum field theory by generating the Feynman diagram expansion of the n-point correlation function with respect to the coupling constant. In this thesis, we give a mathematical formulation of Gell-Mann{ Low formula and the proof of it by using the complex time evolution under certain abstract conditions. Furthermore, we apply the abstract results to QED with cutoffs. (IV) In a Hilbert space, there is a one-to-one correspondence between self-adjoint operators and strongly continuous one-parameter unitary groups. In other words, the self-adjointness of the Hamiltonian is equivalent to the existence of the dynamics in the Schr¨odingerpicture. Furthermore, if the dynamics in the interaction picture exists, then we can translate it into the Schr¨odingerpicture (or the Heisenberg picture). Therefore it is expected that, if the time- ordered exponential representation of the time evolution in the interaction picture exists, then the self-adjointness of the total Hamiltonian follows. Based on this consideration, we present a new theorem concerning a sufficient condition for the essential self-adjointness of a symmetric operator acting in a Hilbert space. By applying the theorem, we prove the essential self-adjointness of the Hamiltonian of the Dirac-Maxwell model in the Coulomb gauge, which is not semi-bounded. 2 Acknowledgements I would like to humbly thank my advisor Asao Arai, for his support, always being kind, encouragement throughout my studies, and having guided me to the exciting world of math- ematical physics. Without his advise and patience this thesis would not have been possible. I have learned many valuable knowledge of mathematics, physics and natural philosophy from him. I would also like to thank Tadahiro Miyao for reading my manuscripts and for his valuable comments. My special words of thanks should also go to my coworker, Kouta Usui. We have had many heated discussions; when I said irrelevant comments, he pointed that out politely. His earnest attitude on research is a model worth imitating, and encouraged me many a time. I could not acquire extremely a lot of important knowledge without his teachings. I would like to thank Akito Suzuki, Toshimitsu Takaesu and Yasumichi Matsuzawa for valuable discussions which formed the basis of some of my research. I express my profound gratitude to members of the Arai research group that have aided me in my growth the past five years. In particular, I mention Daiju Funakawa and Kazuyuki Wada for the various discussions. Finally, and most importantly, I would like to thank my dearest family members, for their understanding, encouragement and unconditional love. My parents, Sohichiroh and Masako, receive my deep gratitude for their dedication. Their support was in the end what made this thesis possible. Most of the results presented in this thesis have been obtained in joint work with Kouta Usui. I would like to thank him for allowing me to make use of those results in this thesis. 3 Contents 1 Introduction 6 1.1 Introduction to time-ordered exponential . 6 1.2 Construction of dynamics for non-symmetric Hamiltonians . 8 1.3 Gupta-Bleuler formalism for the Dirac-Maxwell model . 9 1.4 Gell-Mann{Low formula . 11 1.5 Self-adjointness of Hamiltonian . 12 1.6 A note on notation . 13 2 Abstract construction of time-ordered exponential 13 2.1 Time-ordered exponential on the real axis . 13 2.2 Time-ordered exponential on the complex plane . 22 2.3 Time-ordered exponential as an asymptotic expansion . 36 3 Construction of dynamics for non-symmetric Hamiltonians 39 3.1 Schr¨odingerand Heisenberg equations of motion . 39 3.2 N-derivatives and Taylor expansion . 43 3.3 Application to QED in the Lorenz gauge . 46 3.3.1 Gauge fields . 46 3.3.2 Dirac fields . 50 3.3.3 Total Hamiltonian . 52 3.3.4 η-self-adjointness . 53 3.3.5 Existence of dynamics . 56 4 Gupta-Bleuler formalism for the Dirac-Maxwell model 61 4.1 The Dirac-Maxwell Hamiltonian in the Lorenz gauge . 61 4.1.1 Dirac particle sector . 62 4.1.2 Interaction between the Dirac particles and the gauge field, and the total Hamiltonian . 63 4.2 Time evolution of field operator and field equations . 64 4.3 Current conservation . 66 4.4 Gupta-Bleular condition . 67 5 Gell-Mann { Low formula 74 5.1 Complex time evolution and Gell-Man { Low formula . 74 5.2 Application to QED . 78 5.2.1 Electromagnetic fields . 78 5.2.2 Dirac fields . 80 5.2.3 The total Hamiltonian with cutoffs in the Coulomb gauge . 80 5.2.4 Self-adjointness . 81 5.2.5 Time-ordered exponential on the complex plane . 81 5.2.6 Gell-Mann { Low formula for QED . 83 6 A criteria for essential self-adjointness 88 6.1 Abstract result . 88 6.2 Application to the Dirac-Maxwell Hamiltonian in the Coulomb gauge . 90 7 Future work 93 4 A Fock spaces and second quantizations 93 A.1 Fock spaces . 93 A.2 Second quantization operators . 94 A.3 Boson creation and annihilation operators . 96 A.4 Fermion creation and annihilation operators . 98 B A property of tensor product operator 99 5 1 Introduction In Subsection 1.1, we give a brief and general introduction of time-ordered exponential. Sub- sections 1.2, 1.3, 1.4 and 1.5 are intended to give the relevant background knowledge to Sections 3, 4, 5 and 6, respectively. These subsections require more technical knowledge of mathematics and physics. 1.1 Introduction to time-ordered exponential In this subsection, we give a formal introduction of time-ordered exponential. The rigorous definition and the properties of time-ordered exponential will be discussed in Section 2. Time-ordered exponential is an infinite series defined as follows: Z Z Z n h t io X1 t t 1 0 T exp dτA(τ) := ··· dτ1...dτn T fA(τ1) ··· A(τn)g; t; t 2 R (1.1) 0 n! 0 0 t n=0 t t Here A is a function from R into an algebra. The symbol T is the time-ordering, that is, T denotes the procedure that orders the product A(τ1) ··· A(τn) according to the value of parameter: X T f ··· g ··· A(τ1) A(τn) := χPn (τσ(1); :::; τσ(n))A(τσ(1)) A(τσ(n)); (1.2) σ2Sn where n Pn := f(τ1; :::; τn) 2 R j τ1 > ··· > τng: (1.3) Here Sn denotes the symmetric group of order n, and χJ is the characteristic function of the set J.
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