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ABSTRACT
Two general unsolvability arguments for interacting bosonic quantum field theories are presented, based on Dyson–Schwinger equations on the lattice and cardinality considerations. The first argument is related to the fact that, on a lattice of size N, the system of lattice Dyson–Schwinger equations closes on a basis of ‘primitive correlators’ which is finite, but grows exponentially with N. By properly defining the continuum limit, one finds for a countably infinite basis of the primitive correlators. The second argument is that any conceivable exact analytic calculation of the primitive correlators involves, in the continuum limit, a linear system of coupled partial differential equations on an infinite number of unknown functions, namely the primitive correlators, evolving with respect to an infinite number of independent variables.
Acknowledgments
I wish to express particular thanks to Prof. M. Testa for various discussions. I also acknowledge discussions with Prof. G. Parisi.
Disclosure statement
No potential conflict of interest was reported by the author.
Notes
1 The primitive correlators might equally well be called irreducible correlators (or even master correlators).
2 The existence of the scalar theory in the continuum limit has been proved for d = 2 and d = 3, where the coupling constant λ has a positive mass dimension and the number of primitively divergent diagrams is finite (super-renormalisable cases).
3 If often happens that a system exactly solvable in classical mechanics is also solvable in the quantum theory. Well-known cases are (1) the harmonic oscillator; (2) the Kepler problem, corresponding to the hydrogen atom H; (3) the motion of a particle in the gravitational field of two fixed masses, corresponding to the ion of the molecular hydrogen . Such correspondence is violated in this case: while the free anharmonic oscillator is integrable by quadrature in classical physics (Jacobi elliptic functions are obtained), being an autonomous one-degree of freedom system, the quantum case is not.
4 Correlators involving local composite operators of the form ,
, can be obtained by taking some of the times
equal. Operators containing time derivatives, such as
,
,
, etc., can be obtained by taking time derivatives on both sides of Equation (Equation17
(17) ) and then identifying some of the times.
5 In the Euclidean case, the functional measure
(19)
with
the free (harmonic oscillator) action, is the standard Ornstein–Uhlenbeck measure [Citation1–3].
6 Note the change of sign in the rotation angle with respect to the relation between the corresponding times.
7 That is the dual characterisation or functional characterisation of the circle.
8 It is a ‘compact remnant’ of the symmetry of the theory in (see the previous section).
9 If the theory contains for example a particle with mass , as in our case, that physically means to send to zero the adimensional quantity ma,
(192)
If only massless particles are involved, then one may require for example
(193)
where
is the lowest non-zero energy.
10 For example, at the P+P large hadron collider (LHC), presently operating at the European Center for Nuclear Physics (CERN) at a Centre-of-Mass energy of 13 , processes with up to
partons in the final state are studied.
11 It is like to have an infinite number of distinguishable boxes, but only a finite number of balls to put inside them. Only finitely many boxes are not empty.
12 It is well known that the theory does not exist because the cubic potential is unbounded from below for any real
, so the functional integral is divergent. By giving up unitarity, we can formally overcome this difficulty by taking g purely imaginary.
13 If we were to add to terms linear or cubic in the
's, then odd differences between the
's would appear. The parity of
would not be respected in correlator decomposition.
14 We assume that i and j may coincide (j = i), giving rise in this case to the correlator with ,
.
15 As well known, convergence is exponentially fast in the number k of the digits, as the relative error is ; By adding one digit, one doubles the accuracy.
16 To tame the factorial divergence of the perturbative expansion produced by vacuum instability, one can make the Borel transform of the primitive correlators with respect to , and then write evolution equations in the Borel variables
. This strategy gives rise to a theory similar to the one obtained by direct derivation with respect to the
's.
17 A regular singular point of a second-order ordinary differential equation in λ, say , is a singularity in the equation such that the solution
can be written in a neighbourhood of the origin as the product of a simple function (logarithm, real power, etc.), singular at
, times a convergent power series in λ. For that to occur, the coefficient of the first derivative,
, must contain at most a simple pole at
, while the second derivative
, must contain at most a double pole.
An irregular singular point is a singularity of the equation which is not a regular singular point. The singularity in this case is so strong that it is not anymore possible to write the solution in the factorised form described above.