Foldy-Wouthuysen transformation
Encyclopedia
The Foldy-Wouthuysen transformation (after Lesley L. Foldy and Siegfried A. Wouthuysen) is a unitary transformation on a fermion
wave function of the form:
(1)
where the unitary operator is the 4x4 matrix:
. (2)
Above,
is the unit vector oriented in the direction of the fermion momentum. The above are related to the Dirac matrices by 
and 
, with i=1,2,3. A straightforward series expansion applying the commutativity properties of the Dirac matrices demonstrates that (2) above is true. The inverse 
, so it is clear that 
, where 
is a 4x4 identity matrix.
in bi-unitary fashion, in the form:
(3)
Using the commutativity properties of the Dirac matrices, this can be massaged over into the double-angle expression:
(4)
This factors out into:
(5)
which one chooses. Now comes the distinct question of choosing a particular value for 
, which amounts to choosing a particular transformed representation.
One particularly important representation, is that in which the transformed Hamiltonian operator
is diagonalized. Clearly, a completely diagonalized representation can be obtained by choosing 
such that the 
term in (5) is made to vanish. Such a representation is specified by defining:
(6)
so that (5) is reduced to the diagonalized (this presupposes that
is taken in the Dirac-Pauli representation (after Paul Dirac
and Wolfgang Pauli
) in which it is a diagonal matrix):
(7)
By elementary trigonometry, (6) also implies that:
and 
(8)
so that using (8) in (7) now leads following reduction to:
(9)
This calculation can be examined in further detail in the following link.
Prior to Foldy and Wouthuysen publishing their transformation, it was already known that (9) is the Hamiltonian in the Newton-Wigner (NW) representation (named after Theodore Duddell Newton and Eugene Wigner) of the Dirac equation
. What (9) therefore tells us, is that by applying a FW transformation to the Dirac-Pauli representation of Dirac's equation, and then selecting the continuous transformation paramater
so as to diagonalize the Hamiltonian, one arrives at the NW representation of Dirac's equation, because NW itself already contains the Hamiltonian specified in (9). See this link
If one considers an "on shell" mass -- fermion or otherwise -- given by
, and employs a Minkowski metric tensor for which 
, it should be apparent that the expression 
is equivalent to the 
component of the energy-momentum vector 
, so that (9) is alternatively specified rather simply by 
.
. From (6) or (8), this means that 
, so that 
, and, from (2), that the unitary operator 
. Therefore, any operator 
in the Dirac-Pauli representation upon which we perform a bi-unitary transformation, will be given, for an "at rest" fermion, by:
. (10)
Contrasting the original Dirac-Pauli Hamiltonian Operator
with the NW Hamiltonian (9), we do indeed find the 
"at rest" correspondence:
(11)
with the canonical position operators 
, i.e., we must calculate 
. One good way to approach this calculation, is to start by writing the scalar rest mass 
as 
, and then to mandate that the scalar rest mass commute with the 
. Thus, we may write:
(12)
where we have made use of the Heisenberg canonical commutation relationship
to reduce terms. Then, multiplying from the left by 
and rearranging terms, we arrive at:
(13)
Because the canonical relationship
, the above provides the basis for computing an inherent, non-zero acceleration operator, which specifies the oscillatory motion known as Zitterbewegung
.
deleted (14)
. If we use the result at the very end of section 2 above, 
, then this can be written instead as:
. (15)
Using the above, we need simply to calculate
, then multiply by 
.
The canonical calculation proceeds similarly to the calculation in section 4 above, but because of the square root expression in
, one additional step is required.
First, to accommodate the square root, we will wish to require that the scalar square mass
commute with the canonical coordinates 
, which we write as:
(16)
where we again use the Heisenberg canonical relationship
. Then, we need an expression for 
which will satisfy (16). It is straightforward to verify that:
(17)
will satisfy (16) when again employing
. Now, we simply return the 
factor via (15), to arrive at:
. (18)
This is understood to be the velocity operator in the Newton-Wigner representation. Because:
, (19)
it is commonly thought that the Zitterbewegung
motion arising out of (13), vanishes when a fermion is transformed into the Newton-Wigner representation.
deleted (20)
. Here, (13) remains:
(21)
while (18) becomes:
. (22)
In equation (10) we found that for an "at rest" fermion,
for any operator. One would expect this to include:
, (23)
however, equations (21) and (22) for a
fermion appear to contradict (23).
and rewritten here as:
where
is the 
unit matrix. This Hamiltonian is rewritten, namely divided into two parts:
where
and
where
is the Fine-structure constant
(not to be confused with the Dirac alpha matrices). Letting
into the zero order equation for
and using a particular but known representation of the Dirac operators, yields:
where
are the 
Pauli matrices
. Note that the potential
does not appear in the equation above. The equation for the other spinor is:
where
. Eliminating 
gives:
This is simply the non-relativistic equation for a system with a re-normalized potential and energy eigenvalue:
The higher-order corrections can be obtained by conventional perturbation theory.
This is known as Moore's decoupling technique. Though it resembles the FW transformation, it is computationally and conceptually much simpler. Though misunderstood at first, in part because the fine structure constant
appears in both the equations and the order parameter
requiring care in the "bookkeeping" of the perturbative scheme, Moore's decoupling technique was vindicated for the (relativistic) hydrogen atom
using conventional Rayleigh Schrödinger perturbation theory
and computer algebra and proven to converge to the correct
solution .
It has been applied successfully to relativistic calculations on Alkali metals and represents one of many relativistic perturbative schemes
investigated by Werner Kutzelnigg
.
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
wave function of the form:
(1)where the unitary operator is the 4x4 matrix:
. (2)Above,
is the unit vector oriented in the direction of the fermion momentum. The above are related to the Dirac matrices by and , with i=1,2,3. A straightforward series expansion applying the commutativity properties of the Dirac matrices demonstrates that (2) above is true. The inverse , so it is clear that , where is a 4x4 identity matrix.Foldy-Wouthuysen Transformation of the Dirac Hamiltonian for a Free Fermion
This transformation is of particular interest when applied to the free-fermion Dirac Hamiltonian operator in bi-unitary fashion, in the form: (3)Using the commutativity properties of the Dirac matrices, this can be massaged over into the double-angle expression:
(4)This factors out into:
(5)Choosing a Particular Representation: Newton-Wigner
Clearly, the FW transformation is a continuous transformation, that is, one may employ any value for which one chooses. Now comes the distinct question of choosing a particular value for , which amounts to choosing a particular transformed representation.One particularly important representation, is that in which the transformed Hamiltonian operator
is diagonalized. Clearly, a completely diagonalized representation can be obtained by choosing such that the term in (5) is made to vanish. Such a representation is specified by defining: (6)so that (5) is reduced to the diagonalized (this presupposes that
is taken in the Dirac-Pauli representation (after Paul DiracPaul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
and Wolfgang Pauli
Wolfgang Pauli
Wolfgang Ernst Pauli was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or...
) in which it is a diagonal matrix):
(7)By elementary trigonometry, (6) also implies that:
and (8)so that using (8) in (7) now leads following reduction to:
(9)This calculation can be examined in further detail in the following link.
Prior to Foldy and Wouthuysen publishing their transformation, it was already known that (9) is the Hamiltonian in the Newton-Wigner (NW) representation (named after Theodore Duddell Newton and Eugene Wigner) of the Dirac equation
Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
. What (9) therefore tells us, is that by applying a FW transformation to the Dirac-Pauli representation of Dirac's equation, and then selecting the continuous transformation paramater
so as to diagonalize the Hamiltonian, one arrives at the NW representation of Dirac's equation, because NW itself already contains the Hamiltonian specified in (9). See this linkIf one considers an "on shell" mass -- fermion or otherwise -- given by
, and employs a Minkowski metric tensor for which , it should be apparent that the expression is equivalent to the component of the energy-momentum vector , so that (9) is alternatively specified rather simply by .Correspondence Between the Dirac-Pauli and Newton-Wigner Representations, for an "At Rest" Fermion
Now let us consider a fermion "at rest," which we may define in this context as a fermion for which. From (6) or (8), this means that , so that , and, from (2), that the unitary operator . Therefore, any operator in the Dirac-Pauli representation upon which we perform a bi-unitary transformation, will be given, for an "at rest" fermion, by:. (10)Contrasting the original Dirac-Pauli Hamiltonian Operator
with the NW Hamiltonian (9), we do indeed find the "at rest" correspondence: (11)The Velocity Operator in the Dirac-Pauli Representation
Now, let us consider the velocity operator. To obtain this operator, we must commute the Hamiltonian operator with the canonical position operators , i.e., we must calculate . One good way to approach this calculation, is to start by writing the scalar rest mass as , and then to mandate that the scalar rest mass commute with the . Thus, we may write: (12)where we have made use of the Heisenberg canonical commutation relationship
to reduce terms. Then, multiplying from the left by and rearranging terms, we arrive at: (13)Because the canonical relationship
, the above provides the basis for computing an inherent, non-zero acceleration operator, which specifies the oscillatory motion known as ZitterbewegungZitterbewegung
Zitterbewegung is a theoretical rapid motion of elementary particles, in particular electrons, that obey the Dirac equation...
.
deleted (14)
The Velocity Operator in the Newton-Wigner Representation
In the Newton-Wigner representation, we now wish to calculate. If we use the result at the very end of section 2 above, , then this can be written instead as:. (15)Using the above, we need simply to calculate
, then multiply by .The canonical calculation proceeds similarly to the calculation in section 4 above, but because of the square root expression in
, one additional step is required.First, to accommodate the square root, we will wish to require that the scalar square mass
commute with the canonical coordinates , which we write as: (16)where we again use the Heisenberg canonical relationship
. Then, we need an expression for which will satisfy (16). It is straightforward to verify that: (17)will satisfy (16) when again employing
. Now, we simply return the factor via (15), to arrive at:. (18)This is understood to be the velocity operator in the Newton-Wigner representation. Because:
, (19)it is commonly thought that the Zitterbewegung
Zitterbewegung
Zitterbewegung is a theoretical rapid motion of elementary particles, in particular electrons, that obey the Dirac equation...
motion arising out of (13), vanishes when a fermion is transformed into the Newton-Wigner representation.
deleted (20)
The Velocity Operators for an "At Rest" Fermion
Now, let us compare equations (13) and (18) for a fermion "at rest," defined earlier in section 3 as a fermion for which. Here, (13) remains: (21)while (18) becomes:
. (22)In equation (10) we found that for an "at rest" fermion,
for any operator. One would expect this to include:, (23)however, equations (21) and (22) for a
fermion appear to contradict (23).Similar Alternatives - Perturbative Schemes
Starting with the one-particle Dirac equation written earlier withand rewritten here as:
where
is the unit matrix. This Hamiltonian is rewritten, namely divided into two parts:where
and
where
is the Fine-structure constantFine-structure constant
In physics, the fine-structure constant is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units...
(not to be confused with the Dirac alpha matrices). Letting
into the zero order equation for
and using a particular but known representation of the Dirac operators, yields:where
are the Pauli matricesPauli matrices
The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...
. Note that the potential
does not appear in the equation above. The equation for the other spinor is:where
. Eliminating gives:This is simply the non-relativistic equation for a system with a re-normalized potential and energy eigenvalue:
The higher-order corrections can be obtained by conventional perturbation theory.
This is known as Moore's decoupling technique. Though it resembles the FW transformation, it is computationally and conceptually much simpler. Though misunderstood at first, in part because the fine structure constant
appears in both the equations and the order parameter
requiring care in the "bookkeeping" of the perturbative scheme, Moore's decoupling technique was vindicated for the (relativistic) hydrogen atomHydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...
using conventional Rayleigh Schrödinger perturbation theory
Perturbation theory (quantum mechanics)
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...
and computer algebra and proven to converge to the correct
solution .
It has been applied successfully to relativistic calculations on Alkali metals and represents one of many relativistic perturbative schemes
investigated by Werner Kutzelnigg
Werner Kutzelnigg
Werner Kutzelnigg, born on September 10, 1933 in Vienna, Austria, is a prominent theoretical chemist. He is now emeritus Professor in the Chemistry Faculty, Ruhr-Universität Bochum, Germany....
.