Fock Space

8 Key excerpts on "Fock Space"

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  An Interpretive Introduction to Quantum Field Theory

  • Paul Teller(Author)
  • 2020(Publication Date)
  • Princeton University Press

    (Publisher)

  There is, however, a fact not yet included in our descriptive frame-work. The framework can be used to describe either Bosons or Fermions. Because there can be any number of a kind of Boson, in the Bosonic case we specify that in |ni, ri2 «j, ... , n^)^ each n* can take on any non-negative integer value. Because there can be only zero or one of a given kind of Fermion, when we use this descriptive framework to de-scribe Fermions we specify that in |ni, ni, • • •, ni, . .. ,nk)A each rij can be only either zero or one. As I emphasized in the last chapter, at the level of understanding that I am able to present in this book, this fact is simply written in to correspond to what we observe to be the case in nature. No explanation of this fact is provided or suggested. A Fock Space is now, very simply, a Hilbert space spanned by the vectors n, ni, ..., n,, ..., n^jAi assumed as an orthonormal basis 1 The Fock Space we are developing, with basis vectors n, 712, • • •, ni, ..., n^)^, each representing a finite aggregation of quanta, provides an irreducible representa-tion of the commutation relations introduced in the next section. There are also nonseparable Hilbert spaces with states representing infinite aggregations of quanta which provide representations of the commutation relations. But these larger spaces have no vacuum, that is, no zero-quanta state and no number operator, also to be introduced later. Schweber (1961), pp. 163-64, gives a further summary and ref-erences. Although most practitioners take quantum field theories to employ Fock Space, which is a separable Hilbert space, the complications brought on by Haag's theorem might force the relevance of nonseparable Hilbert spaces, depending on how these complications are resolved. I will very briefly discuss the implications of Haag's theorem in chapter 6. F O C K S P A C E 39 and understood to be generalized on the one-quantum case as I explained earlier.

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  Advanced Quantum Mechanics

  A Practical Guide

  • Yuli V. Nazarov, Jeroen Danon(Authors)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  In the subsequent years, Fock intro- duced the Fock states and Fock Space, which later became fundamental concepts in many-particle theory. In 1930 he also improved Hartree’s method for finding solutions of the many-body Schrödinger equation by including the correct quantum statistics: this method is now know as the Hartree–Fock method (see e.g. Chapter 4). In 1932 Fock was finally appointed professor at the University of Leningrad (as the city was called since 1924). He stayed there until his retirement, making important contributions to a wide range of fields, including general relativity, optics, theory of gravitation, and geophysics. He was brave enough to confront Soviet “philosophers” who stamped the theory of relativity as being bourgeois and idealistic: a stand that could have cost him his freedom and life. system consisting of only a single particle. In the basis of the levels – the allowed single particle states ψ k – we can write any operator acting on a this system as ˆ f =  k,k  f k  k |ψ k  (r)ψ k (r)|, (3.13) where the coefficients f k  k are the matrix elements f k  k = ψ k  (r)| ˆ f |ψ k (r). For a many- particle system, this operator becomes a sum over all particles, ˆ F = N  i=1  k,k  f k  k |ψ k  (r i )ψ k (r i )|, (3.14) where r i now is the coordinate of the ith particle. Since the particles are identical, the coefficients f k  k are equal for all particles. 69 3.2 Field operators for bosons We see that, if in a many-particle state the level k is occupied by n k particles, the term with f k  k in this sum will result in a state with n k − 1 particles in level k and n k  + 1 particles in level k  . In Fock Space we can write for the term f k  k N  i=1 f k  k |ψ k  (r i ) ψ k (r i )| . . . , n k  , . . . , n k , . . . = Cf k  k |. . . , n k  + 1, . . . , n k − 1, . . ., (3.15) where the constant C still has to be determined.

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  Solid State and Quantum Theory for Optoelectronics

  • Michael A. Parker(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  The most important point of the Fock state is that it is an eigenstate of the number operator as we will see. We should include the spin in the description of the Fock state. Assume the spin along the z -direction is represent by s ¼ 1 (up) and s ¼ 2 (down). Each index ~ k value must be augmented with the polarization directions as indicated in Figure 5.78. Therefore, one can create a particle with a given wave vector and given spin. For bosons, which are characterized by integer spin (0, 1, 2, . . . ), any number of them can occupy a mode. For a given set of modes, each Fock state is a basis vector for the amplitude space. The set fj n 1 , n 2 , n 3 , . . . ig represents the complete set of basis vectors where each n i can range up to an in fi nite number of boson particles in the system. The orthonormality relation can be written as h n 1 , n 2 , . . . j m 1 , m 2 , . . . i¼ d n 1 m 1 d n 2 m 2 ( 5 : 353 ) and the closure relation as X 1 n 1 , ; n 2 ... ¼ 0 j n 1 , n 2 . . . ih n 1 , n 2 . . . j¼ ^ 1 ( 5 : 354 ) n 1 = 2 n 3 = 1 n 2 = 0 m = 1 m = 2 m = 3 … | FIGURE 5.77 The Fock state describes the number of particles in the modes or states of the system. The diagram represents the ket j 2, 0, 1, . . . i . s = 1 s = 2 s = 1 s = 2 , , k 2 k 1 … | FIGURE 5.78 The modes must include polarization. Quantum Mechanics 413 A general vector in the Hilbert space must have the form j j i¼ X 1 n 1 , n 2 ... ¼ 0 b n 1 , n 2 ... j n 1 , n 2 . . . i ( 5 : 355 ) where quantum mechanical wave functions must be normalized to unity as usual. The component b n 1 , n 2 ... ¼h n 1 , n 2 , . . . j j i represents the probability amplitude of fi nding n 1 particles in state 1, n 2 particles in state 2, etc. when the system has wave function j j i . Fock states can also be constructed for fermions with half-integral spin, such as electrons with spin ½; however, the Pauli exclusion principle limits the number per mode to at most 1.

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  Introduction to Electrodynamics and Radiation

  • Walter T. Jr. Grandy(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  This is very desirable because, as we h a v e seen, physical processes exist in real systems w h i c h create a n d destroy particles, a n d so one m u s t w o r k with a theory capable of a c c o m m o d a t i n g such interaction terms in the H a m i l -tonian. T h u s , we will n o w generalize the preceding formalism by i n t r o -d u c i n g the n o t i o n of Fock Space? in w h i c h the totality of all TV -body spaces is considered, for TV = 0, 1, 2, . . . . In this space all functions of t h e type (10-18) appear, for all TV. F r o m the last p a r a g r a p h it should be clear that F o c k space is merely t h e direct s u m of all the H u b e r t spaces for TV particles, T V = 0, 1, 2, . . . , a n d can be written symbolically as ^ = Ζ ο θ ^ θ Ζ 2 θ · . (10-21) It follows that one can then define basis vectors in this space in the follow-ing m a n n e r : y N Ξ < χ, · · · x N ι Λ ι . . . W / . . . > l ° 0 ο 1 0 , (10-22) 156 X. The Quantum Mechanics of iV-Particle Systems where the first position in the c o l u m n m a t r i x represents the v a c u u m (a realizable state of the system), a n d the 1 appears in the (TV + l ) t h posi-tion. T h e subscript TV on the Fock vector indicates the n u m b e r of particles in this Fock state, but, because there exists rare cause for confusion, it will generally be omitted. It is to be noted that this generalized space is in-finite dimensional; that is, it contains infinitely m a n y TV-particle subspaces, to allow for creation of a n arbitrary n u m b e r of nonconserved particles. T h e n o r m a l i z a t i o n of the basis vectors is guaranteed by 1 = j I < Xl · · -x N n, · - · n t · · ·> 2 d* Xl · • · d 3 x N9 (10-23) where it is understood that the n o r m a l i z a t i o n of the v a c u u m is <0 I 0> = 1 .

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  Quantum Electrodynamics

  • Eugene Stefanovich(Author)
  • 2018(Publication Date)
  • De Gruyter

    (Publisher)

  1 Fock Space There are more things in Heaven and on earth, dear Horacio, than are dreamed of in your philosophy. Hamlet In this chapter, we construct the Fock Space H populated by particles of five types: electrons e − , positrons e + , protons p + , antiprotons 1 p − and photons γ . We will practice constructions of simple interaction operators and study their properties. In compar-ison with Volume 1, the main novelty is in working with operators that change the number of particles. This will prepare us for mastering a more realistic theory – quan-tum electrodynamics (QED) – in Chapters 3 and 4. 1.1 Creation and annihilation operators Here we introduce the concepts of creation and annihilation operators. Though lack-ing autonomous physical meaning, these operators greatly simplify calculations in H . 1.1.1 Sectors with fixed numbers of particles The numbers of particles of each type are easily measurable in experiments, so we have the right to introduce in our theory five new observables, namely, the numbers of electrons ( N el ), positrons ( N po ), protons ( N pr ), antiprotons ( N an ) and photons ( N ph ). Unlike in ordinary quantum mechanics from Volume 1, here we will not assume that the numbers of particles are fixed. We would like to treat these quantities on the same footing as other quantum observables. In particular, we will also take into account their quantum uncertainty. Then, in accordance with general quantum rules, these observables should be represented in the Hilbert space ( = Fock Space) H by five Her-mitian operators. Obviously, their allowed values (spectra) are nonnegative integers (0 , 1 , 2 , . . . ). From part (II) of Postulate 1 -6.1, it follows that these observables are mea-surable simultaneously, so that the particle number operators commute with each other and have common eigensubspaces.

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  The Quantum Theory of Nonlinear Optics

  • Peter D. Drummond, Mark Hillery(Authors)
  • 2014(Publication Date)
  • Cambridge University Press

    (Publisher)

  We will see later that the field quantization method guarantees this property; in first-quantized methods, it must be imposed by hand. 2.2.1 Many-particle Hilbert space To illustrate the Fock Space formalism, let us recall nonrelativistic quantum mechanics. The Hilbert space of a single particle is H 1 , with an orthonormal basis | f j , where j = 1, 2, . . . . These are often chosen to be the eigenstates of the single-particle Hamiltonian, i.e. h SP | f j  = E j | f j , where in nonrelativistic physics h SP = −¯ h 2 2m ∇ 2 r + V (r). (2.18) 42 Field quantization If we construct the many-body states from single-particle eigenstates, we would use these eigenstates together with symmetrization requirements. We will show that this approach gives the same results as field quantization. Where there are both field and single-particle versions of operators, to distinguish the two types, we use capitals for quantum field operators and lower-case letters for single-particle operators. Because we are now considering bosons, the N -particle Hilbert space will be the sym- metric subspace of the N -fold tensor product of H 1 with itself, i.e. H N = (H ⊗N 1 ) sym . The boson Fock Space, H, is just the direct sum of spaces corresponding to different particle numbers, H = H 0 ⊕ H 1 ⊕ H 2 ⊕ · · · . (2.19) Here, H 0 is the one-dimensional space corresponding to no particles, and we call the normalized vector in it the vacuum state. The Hilbert space H allows us to describe states that do not have a definite number of particles. A state may have a certain amplitude to have no particles, another amplitude to have one particle, and so on. This type of state arises often in quantum optics, where the particles are photons.

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  Statistical Mechanics

  An Intermediate Course

  • G Morandi, E Ercolessi;F Napoli;;(Authors)
  • 2001(Publication Date)
  • WSPC

    (Publisher)

  (5.1.1) is conceptually incorrect and, furthermore: (ii) Eq. (5.1.30) holds (in Euclidean spaces) only for d > 3. Now, Con- densed Matter Physics is more and more concerned nowadays with 296 Identical Particles in Quantum Statistical Mechanics low-dimensional (mainly d — 2) systems like inversion layers in het- erostructures and quantum Hall devices [98; 118], thin 3 He films [130; 154], high-T c superconducting materials [58; 120], to quote only a few of them. So, studying Quantum Mechanics (and Quantum Statistical Mechanics) in d = 2 is far from being an academic exercise. For the rest of this section we will stick to d > 3, where identical particles are honest bosons or fermions, i.e. are polite enough to us to avoid obeying "fancy" statistics like the ones indicated in (5.1.31). 5.2 Fock Spaces & Second Quantization The topic we want to discuss here is a formalism that is particularity convenient for the description of many-particle systems when the particle number is not held constant and hence, typically, one has to do Statisti- cal Mechanics in the grand-canonical ensemble. It goes under the equiv- alent names of "Fock Space" [64], or "second quantization" [5; 47; 104; 135] or "occupation number" representation. We will also consider the case d = 3, although whatever statements we will make here will apply to d > 3 as well. Let's consider spinless particles for the time being, and let {u a (a?)}, a € I with I an index set, be an orthonormal basis of one-particle wave- functions. For example, for free particles in a cubic box of volume V, and assuming periodic boundary conditions, we may take as a basis the eigen- functions of the momentum operator, i.e. the normalized plane waves: u p (x) = -Le i ?-*/ h (5.2.1) with: p = — = n ; n = (n!,n 2 ,n3) ; n f G Z , i = 1,2,3,... (5.2.2) The symmetric group S N will be made to act on K^(E 3N ) in the following manner.

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  Quantum Field Theory And Its Macroscopic Manifestations: Boson Condensation, Ordered Patterns And Topological Defects

  Boson Condensation, Ordered Patterns and Topological Defects

  • Massimo Blasone, Petr Jizba, Giuseppe Vitiello(Authors)
  • 2011(Publication Date)
  • ICP

    (Publisher)

  We will call it the physical Fock Space. We define the free or physical field φ ( x ), with x denoting x ,t , by φ ( x ) = integraldisplay d 3 k bracketleftBig u ( k ) α k e i k · x -iE k t + v ( k ) β † k e -i k · x + iE k t bracketrightBig . (1.131) In general, φ ( x ) is a one-column matrix. The fact that the energy E k of a physical particle is a certain function of its momentum means that the physical field φ ( x ) must solve a linear homogeneous equation: Λ( ∂ ) φ ( x ) = 0 . (1.132) The differential operator Λ( ∂ ) is in general a square matrix. Its operation is defined on the Fourier transform as Λ( ∂ ) e -ik · x = Λ( ik ) e -ik · x , k · x ≡ k μ x μ = E k t -k · x . The “wave functions” u ( k ) and v ( k ) are solutions of Λ( ik ) u ( k ) = 0 , and Λ( -ik ) v ( k ) = 0 , (1.133) respectively. The structure of the space of the physical states 33 Physical particles are thus ingoing and outgoing particles far from the region of interaction. In solid state physics, as mentioned in Section 1.2, the physical particles are called quasiparticles. We will call in-fields or out-fields the fields referring to ingoing or outgoing physical particles, respectively, and denote them by φ in and/or φ out . In the following Chapters, whenever no misunderstanding arises we will drop the ‘in’ and/or ‘out’ indexes. In-fields and out-fields will also be generically called asymptotic fields since they describe particles in spacetime regions where interactions are not felt. The free field equations of type (1.132), in fact, do not contain any in-formation about the interactions. Although the physical particles undergo interaction processes, the language we have set up till now cannot describe such dynamical processes; thus we need another source of information to describe the dynamics of a physical system. The concept of free field is pertinent to one of the aspects of the two-level description of Nature.

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