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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Segal–Bargmann space (for
Irving Segal Irving Ezra Segal (1918–1998) was an American mathematician known for work on theoretical quantum mechanics. He shares credit for what is often referred to as the Segal–Shale–Weil representation. Early in his career Segal became known for h ...
and
Valentine Bargmann Valentine "Valya" Bargmann (April 6, 1908 – July 20, 1989) was a German-American mathematician and theoretical physicist. Biography Born in Berlin, Germany, to a German Jewish The history of the Jews in Germany goes back at least to the y ...
), also known as the Bargmann space or Bargmann–Fock space, is the space of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s ''F'' in ''n'' complex variables satisfying the square-integrability condition: \, F\, ^2 := \pi^ \int_ , F(z), ^2 \exp(-, z, ^2)\,dz < \infty, where here ''dz'' denotes the 2''n''-dimensional Lebesgue measure on \Complex^n. It is a
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
with respect to the associated inner product: \langle F\mid G\rangle = \pi^ \int_ \overlineG(z)\exp(-, z, ^2)\,dz. The space was introduced in the mathematical physics literature separately by Bargmann and Segal in the early 1960s; see and . Basic information about the material in this section may be found in and . Segal worked from the beginning in the infinite-dimensional setting; see and Section 10 of 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 \Complex^n, there is a constant ''C'' such that , F(a), < C\, F\, . It then follows from the
Riesz representation theorem The Riesz representation theorem, sometimes called the Riesz–Fréchet representation theorem after Frigyes Riesz and Maurice René Fréchet, establishes an important connection between a Hilbert space and its continuous dual space. If the un ...
that there exists a unique ''F''''a'' in the Segal–Bargmann space such that F(a) = \langle F_a\mid F\rangle. The function ''F''''a'' may be computed explicitly as F_a(z) = \exp(\overline\cdot z) where, explicitly, \overline\cdot z = \sum_^n \overlinez_j. The function ''F''''a'' is called the
coherent state In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmo ...
(applied in mathematical physics) with parameter ''a'', and the function \kappa(a,z) := \overline is known as the
reproducing kernel In functional analysis, a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H of functions from a set X (to \mathbb or \mathbb) is an ...
for the Segal–Bargmann space. Note that F(a) = \langle F_a\mid F\rangle = \pi^ \int_ \kappa(a,z)F(z)\exp(-, z, ^2)\,dz, meaning that integration against the reproducing kernel simply gives back (i.e., reproduces) the function ''F'', provided, of course that ''F'' is an element of the space (and in particular is holomorphic). Note that \, F_a\, ^2 = \langle F_a\mid F_a\rangle = F_a(a) = \exp(, a, ^2). It follows from the
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...
that elements of the Segal–Bargmann space satisfy the pointwise bounds \left, F(a)\ \leq \left\, F_a\right\, \left\, F\right\, = \exp(, a, ^2/2) \left\, F\right\, .


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^n. In this view, \Complex^n plays the role of the classical phase space, whereas \R^n 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 \pi^ , F(z), ^2 \exp(-, z, ^2) 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 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_j and creation operators a_j^* on the Segal–Bargmann space by setting a_j = \partial /\partial z_j and a_j^* = z_j These operators satisfy the same relations as the usual creation and annihilation operators, namely, the a_j and a_j^* commute among themselves and \left _j, a_k^* \right = \delta_ Furthermore, the adjoint of a_j with respect to the Segal–Bargmann inner product is a_j^*. (This is suggested by the notation, but not at all obvious from the formulas for a_j and a_j^*!) 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''''j'' and ''B''''j'' by the formulas: \begin A_j &= \frac (a_j + a_j^*), & B_j &= \frac (a_j - a_j^*) \end These operators satisfy the ordinary canonical commutation relations, and it can be shown that they act irreducibly on the Segal–Bargmann space; see Section 14.4 of .


The Segal–Bargmann transform

Since the operators and from the previous section satisfy the Weyl relations and act irreducibly on the Segal–Bargmann space, the
Stone–von Neumann theorem In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named aft ...
applies. Thus, there is a unitary map from the position Hilbert space L^2(\R^n) to the Segal–Bargmann space that intertwines these operators with the usual position and momentum operators. The map may be computed explicitly as a modified double
Weierstrass transform In mathematics, the Weierstrass transform of a function f : \mathbb\to \mathbb, named after Karl Weierstrass, is a "smoothed" version of f(x) obtained by averaging the values of f, weighted with a Gaussian centered at x. Specifically, it is the ...
, (Bf)(z) = \int_ \exp \tfrac (z \cdot z - 2 \sqrt z \cdot x + x \cdot x)(x) \, dx, where ''dx'' is the ''n''-dimensional Lebesgue measure on \R^n and where is in \Complex^n. See Bargmann (1961) and Section 14.4 of Hall (2013). One can also describe as the inner product of with an appropriately normalized
coherent state In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmo ...
with parameter , 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 is a unit vector in L^2(\R^n), then we may form a probability density on \Complex^n as \pi^ , (Bf)(z), ^2 \exp(-, z, ^2) ~. The claim is then that the above density is the Husimi function of , which may be obtained from the Wigner function of by convolving with a double Gaussian (the
Weierstrass transform In mathematics, the Weierstrass transform of a function f : \mathbb\to \mathbb, named after Karl Weierstrass, is a "smoothed" version of f(x) obtained by averaging the values of f, weighted with a Gaussian centered at x. Specifically, it is the ...
). This fact is easily verified by using the formula for along with the standard formula for the Husimi function in terms of coherent states. Since is unitary, its Hermitian adjoint is its inverse. Recalling that the measure on \Complex^n is e^\,dz, we thus obtain one inversion formula for as f(x) = \int_ \exp \tfrac(\overline \cdot \overline - 2 \sqrt \overline \cdot x + x \cdot x)Bf)(z) e^\, dz. Since, however, is a holomorphic function, there can be many integrals involving that give the same value. (Think of the Cauchy integral formula.) Thus, there can be many different inversion formulas for the Segal–Bargmann transform . Another useful inversion formula is f(x) = C \exp(-, x, ^2/2) \int_ (Bf)(x+iy)\exp(-, y, ^2/2) \, dy, where C = \pi^ (2\pi)^. This inversion formula may be understood as saying that the position "wave function" may be obtained from the phase-space "wave function" by integrating out the momentum variables. This is to be contrasted to the Wigner function, where the position ''probability density'' is obtained from the phase space (quasi-)''probability density'' by integrating out the momentum variables.


Generalizations

There are various generalizations of the Segal–Bargmann space and transform. In one of these,B.C. Hall,
The inverse Segal–Bargmann transform for compact Lie groups
, ''Journal of Functional Analysis'' 143 (1997), 98–116
the role of the configuration space \R^n is played by the group manifold of a compact
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
, such as SU(''N''). The role of the phase space \Complex^n is then played by the ''complexification'' of the compact Lie group, such as \operatorname(N, \Complex) in the case of SU(''N''). The various Gaussians appearing in the ordinary Segal–Bargmann space and transform are replaced by
heat kernel In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum ...
s. This generalized Segal–Bargmann transform could be applied, for example, to the rotational degrees of freedom of a rigid body, where the configuration space is the compact Lie groups SO(3). This generalized Segal–Bargmann transform gives rise to a system of
coherent state In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmo ...
s, known as heat kernel coherent states. These have been used widely in the literature on
loop quantum gravity Loop quantum gravity (LQG) is a theory of quantum gravity that incorporates matter of the Standard Model into the framework established for the intrinsic quantum gravity case. It is an attempt to develop a quantum theory of gravity based direc ...
.


See also

* Theta representation *
Hardy space In complex analysis, the Hardy spaces (or Hardy classes) H^p are spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In real anal ...


References


Sources

* * * * * * {{DEFAULTSORT:Segal-Bargmann space Function spaces