Non-Hilbert spaces: a way to new physics?
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ELEMENTARY PARTICLES AND FIELDS Theory
Non-Hilbert Spaces: a Way to New Physics?∗ V. V. Belokurov1), 2)** and E. T. Shavgulidze1) Received May 18, 2016
Abstract—We give some examples of the problems where non-Hilbert spaces could be relevant. DOI: 10.1134/S1063778817020065
1. INTRODUCTION In the early years of quantum theory, Dirac tried to make use of the states with the negative value of energy. Now, however, anti-particles are described in the quite different manner. And the eigen-states of quantum harmonic oscillator with negative value of energy are considered as nonphysical ones. The eigen-states of all other Hamiltonians with the wave functions that do not diminish at infinity are supossed to be nonphysical as well (see, e.g. [1]). They do not belong to the Hilbert space and it is unclear how the usual perturbative treatment could be used. Here we discussed several problems with the states that do not belong to the Hilbert space but could have some physical sence (see, also, [2] and [3]). 2. QUANTUM OSCILLATORS First let us consider the one-dimensional Calogero model with the Hamiltonian d2 1 − 2 + x2 + 2x−2 . (1) H= 2 dx There are vast numbers of papers where this problems was studied. As a rule, the wave functions regular at x = 0 are considered. They are expressed in terms of degenerate hypergeometric functions and form a basis in the Hilbert space of functions on the semi-axis 0 < x < +∞ bounded at x = 0. However, there is another class of solutions singular at x = 0. They are generated from the vacuum state 2 x (1) −1 , (2) Ψ0 (x) = x exp − 2 ∗
The text was submitted by the authors in English. Moscow State University, Moscow, 119991 Russia. 2) Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia. ** E-mail: [email protected] 1)
(1)
with the “vacuum” energy E0 = − 12 . All other states have the form + n (1) Ψ0 (x) Ψ(1) n (x) = A 2 −1 x , = x + P2n−1 (x) exp − 2
(3)
(1)
with En = − 12 + 2n. Here, the operator 1 = 2
+ A+ = a+ α a−α
d2 d 2 −2 −1 + x − 2x − 2x dx2 dx
increases the energy by 2 + A , H = ∓2A+ .
(4)
(5)
And P2n−1 (x) is a polynomial with only odd powers of x. From the equation b (1) 2 Ψn (x) dx a = 0, a > 0, (6) lim ∞ ε→0 (1) 2 Ψn (x) dx ε
it follows that the new wave functions are concentrated in the infinitesimal vicinity of x = 0. In spite of the divergency of the integral +∞
x−2 exp(−x2 )dx, 0
it is possible to give a consistent meaning to the norm (1) of the state Ψ0 (x). According to the definition of Γ function at negative arguments, we regularize the divergent integral as +∞
x−2 exp(−x2 )dx 0
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BELOKUROV, SHAVGULIDZE
1 = 2
+∞
3 1 1 t− 2 exp(−t)dt = Γ − 2 2
that commutes with H [Q, H] = 0.
0
and get the norm 1 1 √ (1) (1) = − π. Ψ0 (x), Ψ0 (x) = Γ − 2 2
(7)
In this case, one can easily verify the orthogonality (1) of the functions Ψn (1) (x), Ψ (x) = 0, Ψ(1) n m n, m = 0, 1, . . . ,
n = m,
and the self-adjointness of the operator H. (1)
Although the norm of Ψ0 is negative and in this
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