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In [[mathematics]], the '''Segal–Bargmann space''' (for [[Irving Segal]] and [[Valentine Bargmann]]), also known as the '''Bargmann space''' or '''Bargmann–Fock space''', is the space of [[holomorphic]] functions ''F'' in ''n'' complex variables satisfying the square-integrability condition:
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:<math>\|F\|^2 := \pi^{-n} \int_{C^n} |F(z)|^2 \exp(-|z|^2)\,dz < \infty,</math>
 
where here ''dz'' denotes the 2''n''-dimensional Lebesgue measure on ''C''<sup>''n''</sup>. It is a Hilbert space with respect to the associated inner product:
:<math> \langle F\mid G\rangle = \pi^{-n} \int_{C^n} \overline{F(z)}G(z)\exp(-|z|^2)\,dz. </math>
 
The space was introduced in the mathematical physics literature separately by Bargmann and Segal in the early 1960s; see {{harvtxt|Bargmann|1961}} and {{harvtxt|Segal|1963}}. Basic information about the material in this section may be found in {{harvtxt|Folland|1989}} and {{harvtxt|Hall|2000}}. Segal worked from the beginning in the infinite-dimensional setting; see {{harvtxt|Baez|Segal|Zhou|1992}} and Section 10 of {{harvtxt|Hall|2000}} for more information on this aspect of the subject.
 
== Properties ==
 
A basic property of this space is that ''pointwise evaluation is continuous'', meaning that for each ''a'' in ''C''<sup>''n''</sup>, there is a constant ''C'' such that
 
:<math> |F(a)|< C\|F\|. </math>
 
It then follows from the [[Riesz representation theorem]] that there exists a unique ''F''<sub>''a''</sub> in the Segal–Bargmann space such that
 
:<math> F(a) = \langle F_a\mid F\rangle. </math>
 
The function ''F''<sub>''a''</sub> may be computed explicitly as
:<math> F_a(z) = \exp(\overline{a}\cdot z) </math>
where, explicitly,
 
:<math> \overline{a}\cdot z = \sum_{j=1}^n \overline{a_j}z_j. </math>
 
The function ''F''<sub>''a''</sub> is called the '''[[coherent state]]''' with parameter ''a'', and the function
:<math> \kappa(a,z) := \overline{F_a(z)} </math>
is known as the '''[[reproducing kernel]]''' for the Segal–Bargmann space. Note that
 
: <math> F(a) = \langle F_a\mid F\rangle = \pi^{-n} \int_{C^n} \kappa(a,z)F(z)\exp(-|z|^2)\,dz,</math>
 
meaning that integration against the reproducing kernel simply gives back (i.e., reproduces) the function ''F'', provided, of course that ''F'' is holomorphic!
 
Note that
 
:<math> \|F_a\|^2 = \langle F_a\mid F_a\rangle = F_a(a) = \exp(|a|^2). </math>
 
It follows from the [[Cauchy–Schwarz inequality]] that elements of the Segal–Bargmann space satisfy the pointwise bounds
 
:<math> |F(a)| \leq \|F_a\| \|F\| = \exp(|a|^2/2)\|F\|. </math>
 
== Quantum mechanical interpretation ==
 
One may interpret a unit vector in the Segal–Bargmann space as the wave function for a quantum particle moving in ''R''<sup>''n''</sup>. In this view, ''C''<sup>''n''</sup> plays the role of the classical phase space, whereas ''R''<sup>''n''</sup> is the configuration space. The restriction that ''F'' be holomorphic is essential to this interpretation; if ''F'' were an arbitrary square-integrable function, it could be localized into an arbitrarily small region of the phase space, which would go against the uncertainty principle. Since, however, ''F'' is required to be holomorphic, it satisfies the pointwise bounds described above, which provides a limit on how concentrated ''F'' can be in any region of phase space.
 
Given a unit vector ''F'' in the Segal–Bargmann space, the quantity
 
:<math>\pi^{-n}|F(z)|^2 \exp(-|z|^2)</math>
 
may be interpreted as a sort of phase space probability density for the particle. Since the above quantity is manifestly non-negative, it cannot coincide with the [[Wigner function]] of the particle, which usually has some negative values. In fact, the above density coincides with the [[Husimi Q representation|Husimi function]] of the particle, which is obtained from the Wigner function by smearing with a Gaussian. This connection will be made more precise below, after we introduce the Segal–Bargmann transform.
 
== The canonical commutation relations ==
 
One may introduce '''annihilation operators''' ''a''<sub>''j''</sub> and '''creation operators''' ''a''<sub>''j''</sub><sup>*</sup>  on the Segal–Bargann space by setting
 
:<math> a_j = \partial /\partial z_j </math>
 
and
 
:<math> a_j^* = z_j </math>
These operators satisfy the same relations as the usual creation and annihilation operators, namely, the ''a''<sub>''j''</sub>'s and ''a''<sub>''j''</sub><sup>*</sup>'s commute among themselves and
 
:<math> [a_j,a_k^*] = \delta_{j,k} </math>
 
Furthermore, the adjoint of ''a''<sub>''j''</sub> with respect to the inner product in (x) is ''a''<sub>''j''</sub><sup>*</sup>. (This is suggested by the notation, but not at all obvious from the formulas for ''a''<sub>''j''</sub> and ''a''<sub>''j''</sub><sup>*</sup>!) Indeed, Bargmann was led to introduce the particular form of the inner product on the Segal–Bargmann space precisely so that the creation and annihilation operators would be adjoints of each other.
 
We may now construct self-adjoint "position" and "momentum" operators ''A''<sub>''j''</sub> and ''B''<sub>''j''</sub> by the formulas:
 
: <math> A_j = (a_j+a_j^*)/2 </math>
: <math> B_j = (a_j - a_j^*)/(2i) </math>
 
These operators satisfy the ordinary canonical commutation relations. It can be shown that ''A''<sub>''j''</sub> and ''B''<sub>''j''</sub> satisfy the exponentiated commutation relations (i.e., the [[Stone–von Neumann theorem|Weyl relations]]) and that they act irreducibly on the Segal–Bargmann space; see Section 14.4 of {{harvtxt|Hall|2013}}.
 
== The Segal–Bargmann transform ==
 
Since the operators ''A''<sub>''j''</sub> and ''B''<sub>''j''</sub> from the previous section satisfy the Weyl relations and act irreducibly on the Segal–Bargmann space, the [[Stone–von Neumann theorem]] applies. Thus, there is a unitary map ''B'' from the position Hilbert space ''L''<sup>2</sup>(''R''<sup>''n''</sup>) to the Segal–Bargmann space that intertwines these operators with the usual position and momentum operators. The map ''B'' may be computed explicitly as
 
:<math> (Bf)(z) = \int_{R^n} \exp[-(z \cdot z - 2 \sqrt{2} z \cdot x + x \cdot x)/2]f(x) \, dx,  </math>
 
where d''x'' is the ''n''-dimensional Lebesgue measure on ''R''<sup>''n''</sup> and where z is in ''C''<sup>''n''</sup>. See Bargmann (1961) and Section 14.4 of Hall (2013). One can also describe (''Bf'')(''z'') as the inner product of ''f'' with an appropriately normalized [[coherent state]] with parameter ''z'', where now we express the coherent states in the position representation instead of in the Segal–Bargmann space.
 
We may now be more precise about the connection between the Segal–Bargmann space and the Husimi function of a particle. If ''f'' is a unit vector in L^2(R^n), then we may form a probability density on C^n as
 
:<math> \pi^{-n} |(Bf)(z)|^2 \exp(-|z|^2),\, </math>
 
The claim is then that the above density is the [[Husimi Q representation|Husimi function]] of ''f'', which may be obtained from the [[Wigner function]] of ''f'' by convolving with a Gaussian. This fact is easily verified by using the formula for ''Bf'' along with the standard formula for the [[Husimi Q representation|Husimi function]] in terms of coherent states. See also Section 8 of Hall (2000).
 
Since ''B'' is unitary, its Hermitian adjoint is its inverse. We thus obtain one inversion formula for ''B'' as
 
:<math> f(x) = \int_{C^n} \exp[-(\overline{z} \cdot \overline{z} - 2 \sqrt{2} \overline{z} \cdot x + x \cdot x)/2](Bf)(z) \, dz,  </math>
 
Since, however, ''Bf'' is a holomorphic function, there can be many integrals involving ''Bf'' that give the same value. (Think of the Cauchy integral formula.) Thus, there can be many different inversion formulas for the Segal–Bargmann transform ''B''. Another useful inversion formula is<ref>B.C. Hall, "The range of the heat operator", in ''The Ubiquitous Heat Kernel'', edited by Jay Jorgensen and Lynne Walling, AMS 2006, pp. 203–231</ref>
:<math> f(x) = C \exp(-|x|^2/2) \int_{R^n} (Bf)(x+iy)\exp(-|y|^2/2) \, dy,  </math>
 
where <math> C = \pi^{-n/4} (2\pi)^{-n/2}</math>. This inversion formula may be understood as saying that the position ''wave function'' ''f'' may be obtained from the phase space ''wave function'' ''Bf'' by integrating out the momentum variables. This should be contrasted to the situation with the Wigner function, where the position ''probability density'' is obtained from the phase space (pseudo-) ''probability density'' by integrating out the momentum variables.
 
== Generalizations ==
 
There are various generalizations of the Segal–Bargmann transform. In one of these,<ref>B.C. Hall, "The Segal–Bargmann 'coherent state' transform for compact Lie groups", ''J. Functional Analysis'' '''122''' (1994), 103–151</ref><ref>B.C. Hall, "The inverse Segal–Bargmann transform for compact Lie groups", ''J. Functional Analysis'' '''143''' (1997), 98–116</ref> the role of the configuration space ''R''<sup>''n''</sup> is played by the group manifold of a compact Lie group, such as SU(''N''). The role of the phase space ''C''<sup>''n''</sup> is then played by the ''complexification'' of the compact Lie group, such as SL(''N'';''C'') in the case of SU(''N''). The various Gaussians appearing in the ordinary Segal–Bargmann space and transform are replaced by [[heat kernel]]s. See Olafsson (2014) for more information.
 
==See also==
* [[Theta representation]]
* [[Hardy space]]
 
== References ==
{{reflist}}
 
== Sources ==
{{Refbegin}}
* {{Citation | first = V. | last = Bargmann | title = On a Hilbert space of analytic functions and an associated integral transform | journal= Communications on Pure and Applied Mathematics | volume = 14 | year=1961 | page = 187 | doi = 10.1002/cpa.3160140303 | issue = 3 }}
* {{citation|first = I. E. | last = Segal | year = 1963 | contribution = Mathematical problems of relativistic physics | at = Chap. VI | title = Proceedings of the Summer Seminar, Boulder, Colorado, 1960, Vol. II | editor-first = M. | editor-last = Kac | series = Lectures in Applied Mathematics | publisher = American Mathematical Society }}
* {{Citation | first = G. | last = Folland | title = Harmonic Analysis in Phase Space | publisher=Princeton University Press | year=1989 }}
* {{Citation | first1 = J. | last1 = Baez | first2 = I. E. | last2 = Segal | author3 = | first3 = Z. | last3 =  Zhou | title = Introduction to Algebraic and Constructive Quantum Field Theory | publisher=Princeton University Press | year=1992 }}
* Hall, B. C. (2000), "Holomorphic methods in analysis and mathematical physics", in ''First Summer School in Analysis and Mathematical Physics'' (S. Pérez-Esteva and C. Villegas-Blas, Eds.), 1–59, Contemporary Mathematics '''260''', Amer. Math. Soc.
* {{Citation | first = B. C. | last = Hall | title = Quantum Theory for Mathematicians | publisher=Springer | year=2013 }}
* {{Citation | first = G. | last = Olafsson | title = The Segal–Bargmann Transform on Euclidean Space and Generalizations | publisher=World Scientific | year=2014 }}
{{Refend}}
 
{{DEFAULTSORT:Segal-Bargmann space}}
[[Category:Mathematics]]

Revision as of 00:13, 2 March 2014

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