2 A Comment on the Question of Degeneracies in Quantum Mechanics
43
second quantization of the B-O equation (2.5), i.e. T N + E NN + E e . This means
that whenever the system is in a degenerate situation in a quantum mechanical description, according to Eq. (2.2), it is simultaneously degenerate in the quantum field
description, see to Eq. (2.6). What is indeed striking is the way the quantum mechanical approach on the one hand and the field theoretical approach on the other hand,
in reality are capable to eliminate the degeneracy. The mechanical approach leads to
multiconfigurational interaction descriptions including all intersecting potential surfaces, as is standard practice in the theory of J-T effect [3, 4], while the field theoretical approach uses the Fröhlich transformation [13] (which, unfortunately, is unable
to remove the degeneracies) and subsequently the Bogoljubov-Valatin transformation [5], the latter reflecting the BCS theory [6], as is customary in the theory of
superconductivity. Both approaches lead to broken symmetries. Again we find here
a striking difference: the mechanical approach of the J-T effect leads to a structural
symmetry breaking, whereas symmetry violations in the field theoretical approach
to BCS theory relates to charge superselection rule violation.
At this point, we may correctly speculate over the differences between the two
approaches, quantum mechanical and quantum field theoretical ones, and in particular over the origin of these differences as, e.g., resulting from the B-O approximation, and moreover how to completely bypass this almost undefeatable approximation. It provides a certain type of virtual degeneracies, and therefore the question
appears whether these are still justified and if there exists some higher principle,
that would entirely circumvent such circumstances and arrive without more ado at
the desired lifting of the degeneration.
Actually, there is an, in principle, exact formulation in quantum mechanics, considered by Monkhorst [7, 8], which ignores the B-O approximation, but, however,
suffers some disadvantages. Firstly, it is not possible to derive analytic expressions
for quantum mechanical measurable quantities, cf. the B-O separation procedure;
and secondly, even with the best computers the computations are numerically exceedingly demanding. It is in effect prohibiting even going beyond such a humble endeavour as just about ten considered particles, electrons and nuclei included.
Hence, unfortunately, it is quite impossible to consider systems where the B-O approximation leads to electronically degenerated states, such as those leading to the
J-T effect or the mechanisms of superconductors.
Regarding quantum field methods, there do not seem to exist any definite techniques in consideration of how to construct a field that is not based on the B-O approximation, or in other words going further than the model Hamiltonian (2.6). This
Hamiltonian representation has turned out to be especially advantageous in treating systems like insulators and conductors, but alas in superconductivity it points
to the same problems in analogy with non-adiabatic corrections in the J-T problem,
i.e. one obtains B-O degenerate states that we then try to eliminate in a subsequent
treatment.
The primary problem of the B-O approximation is related to the centre-of-mass
(COM) notion. It was indeed one of the main reasons why Monkhorst promoted his
concept and entirely avoided to make this approximation. However, it also appears
that the mistake to determine the centre of gravity in the B-O approximation may be
43
second quantization of the B-O equation (2.5), i.e. T N + E NN + E e . This means
that whenever the system is in a degenerate situation in a quantum mechanical description, according to Eq. (2.2), it is simultaneously degenerate in the quantum field
description, see to Eq. (2.6). What is indeed striking is the way the quantum mechanical approach on the one hand and the field theoretical approach on the other hand,
in reality are capable to eliminate the degeneracy. The mechanical approach leads to
multiconfigurational interaction descriptions including all intersecting potential surfaces, as is standard practice in the theory of J-T effect [3, 4], while the field theoretical approach uses the Fröhlich transformation [13] (which, unfortunately, is unable
to remove the degeneracies) and subsequently the Bogoljubov-Valatin transformation [5], the latter reflecting the BCS theory [6], as is customary in the theory of
superconductivity. Both approaches lead to broken symmetries. Again we find here
a striking difference: the mechanical approach of the J-T effect leads to a structural
symmetry breaking, whereas symmetry violations in the field theoretical approach
to BCS theory relates to charge superselection rule violation.
At this point, we may correctly speculate over the differences between the two
approaches, quantum mechanical and quantum field theoretical ones, and in particular over the origin of these differences as, e.g., resulting from the B-O approximation, and moreover how to completely bypass this almost undefeatable approximation. It provides a certain type of virtual degeneracies, and therefore the question
appears whether these are still justified and if there exists some higher principle,
that would entirely circumvent such circumstances and arrive without more ado at
the desired lifting of the degeneration.
Actually, there is an, in principle, exact formulation in quantum mechanics, considered by Monkhorst [7, 8], which ignores the B-O approximation, but, however,
suffers some disadvantages. Firstly, it is not possible to derive analytic expressions
for quantum mechanical measurable quantities, cf. the B-O separation procedure;
and secondly, even with the best computers the computations are numerically exceedingly demanding. It is in effect prohibiting even going beyond such a humble endeavour as just about ten considered particles, electrons and nuclei included.
Hence, unfortunately, it is quite impossible to consider systems where the B-O approximation leads to electronically degenerated states, such as those leading to the
J-T effect or the mechanisms of superconductors.
Regarding quantum field methods, there do not seem to exist any definite techniques in consideration of how to construct a field that is not based on the B-O approximation, or in other words going further than the model Hamiltonian (2.6). This
Hamiltonian representation has turned out to be especially advantageous in treating systems like insulators and conductors, but alas in superconductivity it points
to the same problems in analogy with non-adiabatic corrections in the J-T problem,
i.e. one obtains B-O degenerate states that we then try to eliminate in a subsequent
treatment.
The primary problem of the B-O approximation is related to the centre-of-mass
(COM) notion. It was indeed one of the main reasons why Monkhorst promoted his
concept and entirely avoided to make this approximation. However, it also appears
that the mistake to determine the centre of gravity in the B-O approximation may be