By Polyakov A.

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**Sample text**

104) λ where ψλ (x) are the orthonormal eigenstates of a single-particle Hamiltonian. Instead of characterizing the quantum statistics of a collection of fermions in terms of the antisymmetry of their state vectors, which as we have seen is a bit messy or at least requires a substantial amount of indices-writing, it is now taken care of by the simple anti-commutation relations for the creation and annihilation operators. The price paid for this enormous simpliﬁcation is of course that the operators now are operators on a super-space, the multi-particle space.

The combinatorics, works for any set of one-particle states, say labeled by μ, so that corresponding to the one-particle operator |μ2 μ1 | corresponds the operator F (1) in the multi-particle space F (1) = a†μ2 aμ1 . 12) λ,λ and by linearity the corresponding operator F (1) in the multi-particle space is thus λ|fˆ(1) |λ a†λ aλ . 13) λ,λ We note that if fˆ(1) is hermitian in the one-particle state space, as is F (1) in the multi-particle state space. The total momentum operator P in the multi-particle space is thus P = dp p a†p ap = dx ψ † (x) i ∇x ψ(x) expressed in either the momentum or position representation of the ﬁeld.

21) p where p = p2 /2m is the kinetic energy of the free particle with momentum p. The sum over momenta occurs, one momentum state per momentum volume Δp = (2π )3 /V in three dimensions, as the particles are assumed enclosed in a box of volume V . 2. 3. e. 22) where pF is the Fermi momentum, the radius of the sphere of occupied momentum states (in three dimensions pF = (3π 2 n)1/3 , where n = N/V is the density of the electrons). 4. Show that the vacuum state is non-degenerate and uniquely characterized by all the eigenvalues of the state number operators np being zero.

### 2D Quantum Gravity and SC at high Tc by Polyakov A.

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