this eigenstate. Any state could be represented as a linear combination of eigenstates of energy; for example, , where are constant coefficients. As in classical mechanics, all dynamical operators can be represented by functions of the position and momentum ones, and , respectively. The connection between this representation and the more usual wavefunction representation is given by the eigenstate of the position operator representing a particle at position , which is denoted by an element in the Hilbert space, and which satisfies . Then, . Likewise, the eigenstates of the momentum operator specify the momentum representation: . The central relation between these operators is a quantum analog of the above Poisson bracket of classical mechanics, the canonical commutation relation, . This relation encodes (and formally leads to) the uncertainty principle, in the form . This algebraic structure may be thus considered as the quantum analog of the canonical structure of classical mechanics. Many-particle systems When turning to N-particle systems, i.e., systems containing N identical particles (particles characterized by the same quantum numbers such as mass, charge and spin), it is necessary to extend the single-particle state function to the N-particle state function . A fundamental difference between classical and quantum mechanics concerns the concept of indistinguishability of identical particles. Only two species of particles are thus possible in quantum physics, the so-called bosons and fermions which obey the rules: (bosons), (fermions). Where we have interchanged two coordinates of the state function. The usual wave function is obtained using the Slater determinant and the identical particles theory. Using this basis, it is possible to solve various many-particle problems. Issues and limitations Classical and quantum brackets Dirac's book details his popular rule of supplanting Poisson brackets by commutators: One might interpret this proposal as saying that we should seek a "quantization map" mapping a function on the classical phase space to an operator on the quantum Hilbert space such that It is now known that there is no reasonable such quantization map satisfying the above identity exactly for all functions and . Groenewold's theorem One concrete version of the above impossibility claim is Groenewold's theorem (after Dutch theoretical physicist Hilbrand J. Groenewold), which we describe for a system with one degree of freedom for simplicity. Let us accept the following "ground rules" for the map . First, should send the constant function 1 to the identity operator. Second, should take and to