The Faddeev-Popov Method Demystified

Marco Ornigotti
Institute of Applied Physics, Friedrich-Schiller University,
Jena, Max-Wien Platz 1, 07743 Jena, Germany∗

Andrea Aiello
Max Planck Institute for the Science of Light, Günther-Scharowsky-Strasse 1/Bau24, 91058 Erlangen, Germany and
Institute for Optics, Information and Photonics,
University of Erlangen-Nuernberg, Staudtstrasse 7/B2, 91058 Erlangen, Germany
(Dated: August 5, 2014)
We discuss how to implement the Legendre transform using the Faddeev-Popov method of Quan-
tum Field Theory. By doing this, we provide an alternative way to understand the essence of the
Faddeev-Popov method, using only concepts that are very familiar to the students (such as the
Legendre transform), and without needing any reference to Quantum Field Theory. Two examples
of Legendre transform calculated with the Faddeev-Popov method are given to better clarify the
point.

I. INTRODUCTION understanding for the students (either graduate or un-

dergraduate) that encounter it for the first time, and the
Students learn in the early days of their studies that development of this method reported in standard quan-
the electromagnetic field can be described by a single tum field theory books [4–6] is not helpful for having a
quantity, the vector potential Aµ , instead of the sepa- better understanding of it. In fact, most of the times it
rate electric and magnetic fields [1]. However, two dis- appears more like a magic trick rather than motivated by
tinct vector potentials Aµ and A0µ = Aµ + ∂µ Λ(x) (where some physical argument.
Λ(x) is a scalar field, the gauge field, that satisfies the The Legendre transform, on the other hand, is a very
Helmholtz equation) generate the same electric and mag- simple mathematical instrument that a student knows
netic fields. In order to remove this ambiguity, one should very well, since it finds application in a wide variety of
first fix the gauge, namely determine uniquely the gauge physical problems, like the transformation between the
function Λ(x). Once the gauge has been chosen, the cor- Hamiltonian and the Lagrangian function of a system in
respondence between the vector potential and the elec- classical mechanics [7], the analysis of equilibrium states
tromagnetic fields is made unique. The electromagnetic in statistical physics [8] or the transformation between
field, however, is not the only gauge field in Nature. In the thermodynamical functions (entropy, enthalpy, Gibbs
fact, all the fundamental forces, such as the weak force free energy, etc.) [9]. This mathematical tool has the ad-
(responsible for radioactive decay and nuclear fusion of vantage, with respect to the Faddeev-Popov method, to
subatomic particles) and the strong force (that ensures be of immediate understanding, because it involves a very
the stability of ordinary matter) are both described by well known procedure that any student has seen at least
gauge fields [2]. one time during his studies. The possibility of illustrat-
Since the quantum theory of a gauge field is often de- ing the results of the Faddeev-Popov “magic” method in
scribed by means of the path integral method [3], a gener- terms of a Legendre transform would be of great didac-
alization of the concept of gauge fixing within this frame- tical insight, since it will allow a better understanding of
work is of paramount importance for the development of the physical foundations of a not so easy to understand
the theory itself. The Faddeev-Popov method exactly common method in quantum field theory.
complies with this needing. From a rigorous point of It is then the aim of this paper to unravel this connec-
view, this method consists in using a gauge fixing condi- tion, by using a simple example in which the Faddeev-
tion to reduce the number of allowed orbits of the math- Popov method is used to implement the Legendre trans-
ematical configurations that represent a given physical form of a given function f (x). This is possible because
system to a smaller set, where all the orbits are related by these two methods, apparently very different and far
a smaller gauge group symmetry [2]. In simpler words, away from each other, share in reality the same essence,
the Faddeev-Popov method consists in applying a con- namely they both consist in applying constraint to cer-
straint to the considered field, that automatically imple- tain physical systems: the Faddeev-Popov method fixes
ments the gauge fixing condition, thus determining the the field gauge by imposing the gauge condition as a con-
field unambiguously [5]. straint to the field, and the Legendre transform imple-
Usually, this method is object of doubts and of difficult ments the transformation between a function f (x) and
its transformed pair g(p) by imposing a certain constraint
to the function F (x, p) = xp − f (x) [11].
This work is organized as follows: in Sect. II we re-
∗ Electronic address: marco.ornigotti@uni-jena.de view very briefly the Faddeev-Popov method, in the sim-
 2

plest case of the electromagnetic field. This allows us In order to fully understand the essence of this method,
to write Eq. (8), that constitutes central point for our let us consider the following resolution of the identity [4]:
work. In Sect. III, a brief recall on the definition of !
δG[AΛ ] 
Z
Legendre transform is given. Finally, in Sect. IV we de-
rive the Legendre transform formula by making use of I = det DΛδ(G[AΛ ]), (5)
δΛ


the Faddeev-Popov rule. Conclusions are then drawn in Λ=0

Sect. V. where the derivative in round brackets has to be intended

in the sense of functional derivative [5]. The Dirac delta
function inside the integral is nothing else but the gauge
II. THE FADDEEV-POPOV METHOD IN A condition that one needs to impose to the Lagrangian
NUTSHELL
of the free electromagnetic field [10] in order to fix the
gauge, namely
One of the possible ways to quantize a field is the so-
called path integral quantization. In this scheme, the
transition probability for a physical system to evolve 1
G[AΛ ] = ∂ µ Aµ + ∂ µ ∂µ Λ(x), (6)
from an initial configuration φ(xi ) to a final configuration e
φ(xf ) is obtained by summing all the probability ampli- where e is the electron charge. Eq. (5) is simply a gener-
tudes corresponding to all possible paths in space-time alization to the domain of functional analysis of the well
that the system will take to reach the final point xf start- known expression [14]
ing from the initial point xi [12]. Then, all the transition
amplitudes (n-point functions) can be obtained from a "  #−1 Z
∂g(x) 
generating functional (the path integral) of the form 1= dx δ (g(x)) , (7)
∂x


Z R 4 x=x0
µ
Z[J] = DAµ ei d x (L+J Aµ ) , (1)
where x0 is the point at which g(x) vanishes. Inserting
the expansion of the unity as given by Eq. (5) into Eq.
where L ≡ L[Aµ , ∂ν Aµ , x] is the Lagrangian density, Aµ
(1) (and dropping the bar symbol from Āµ ) gives:
represent the field variables and J µ is an external current.
For the free electromagnetic field the Lagrangian density
Z Z R 4 µ

is given by L = −(1/4)Fµν F µν , and since there are no Z[J] = DΛ DAµ ∆[A]δ(G[AΛ ])ei d x(L+J Aµ ) ,
field sources, J µ has to be set to zero at the end of the (8)
calculation [13]. where
Following Sect. 7.2 of Ref. [4], we can introduce heuris- !
tically the Faddeev-Popov method by making the fol- δG[AΛ ] 
∆[A] = det . (9)

lowing observations. First of all, the integral measure δΛ


DAµ = Πx dAµ (x) in Eq. (1) accounts for every pos- Λ=0

sible Aµ , thus including those that are connected via a This result should be compared with Eq. (4). In this
gauge transformation. We can make this fact explicit by case, in fact, the generating functional Z[J] is regular-
introducing the following notation: ized, thanks to the presence of the Dirac delta function,
Aµ → ĀΛ whose role is to uniquely determine the vector potential
µ. (2)
Aµ by suitably fixing the gauge (6) that is given as the
This means that now we are considering Aµ as a rep- argument of the Dirac delta function itself. This is the
resentative of the class of vector potentials that can be essence of the Faddeev-Popov method: by inserting in
obtained from a given Āµ by performing a gauge trans- the generating functional (1) the resolution of the iden-
formation of the type tity as given by Eq. (5), one is now able to introduce
a constraint (through a Dirac delta function) that auto-
Aµ = Āµ + ∂µ Λ(x). (3) matically fixes the gauge, thus eliminating the arbitrari-
ness of the vector potential Aµ and removing the problem
With this trick, we can rewrite the generating functional
of overcounting the field configurations.
Z[J] as follows, by separating the contributions of Āµ
Let us now put this result in a more appealing form,
and Λ(x):
that will be helpful for the results of the next section. To
this aim, let us rewrite Eq. (8) in the following way:
Z R 4
Z
µ
Z[J] = DĀµ ei d x (L+J Āµ ) DΛ. (4) Z
R Z[J] = DΛ F[Aµ , Λ; J]∆[Aµ ]δ(G[AΛ ]), (10)
The presence of the second integral DΛ is the one that
causes Z[J] to diverge, since there are infinitely many
where
gauge fields Λ(x) that satisfy Eq. (3). The regulariza- Z
tion of the integral (4) is the scope of the Faddeev-Popov R
d4 x(L+J µ Aµ )
method. F[Aµ , Λ; J] = DAµ ei , (11)
 3

and the dependence of F on Λ is implicitly contained in IV. IMPLEMENTING THE LEGENDRE

Aµ through Eq. (3). Although at a first glance Eq. (10) TRANSFORM WITH THE FADDEEV-POPOV
may appear cumbersome and of little use, its form has METHOD
a great advantage: it may be directly used to calculate
the Legendre transform of an arbitrary function f (x), We now want to show that another way of obtain-
as we will discuss in the next section. Moreover, written ing the Legendre transform is by applying the Faddeev-
in this form, the physical idea behind the Faddeev-Popov Popov trick to F (x, p) as given by Eq. (10). To do this,
method appear more clear: given a functional F[Aµ , Λ; J] we firstly rewrite Eq. (10) for the case of ordinary func-
that depends on the particular choice of the gauge func- tions (instead of functionals) as follows:
tion Λ(x), the Faddeev-Popov method consists in insert- Z
ing the gauge fixing condition G[AΛ ] in the form of an
 
I= dx F (x, p)∆(p, x0 )δ G(x, p) , (14)
integral over the gauge configurations Λ. The presence of
the term ∆[Aµ , 0] simply accounts for the correct normal-
ization term to insert in order not to modify the physics where x0 is the point in which G(x, p) = 0 with respect
of the original system. to x and ∆(p, x0 ) is a function to be yet determined.
For the purpose of this note, we will assume that the
function G(x, p) vanishes only for a single given x0 in
the considered interval. We can expand the Dirac delta
function in the integral with the usual expansion formula
III. THE LEGENDRE TRANSFORM
[14]

δ(x − x0 )
δ(w(x)) = , (15)
Before calculating the Legendre transform by means |w0 (x0 )|
of the Faddeev-Popov method, let us first briefly recall
the definition of Legendre transform and clarify its use where x0 is the zero of the function w(x) and w0 (x0 ) is the
with a simple example. Following the standard textbook first derivative of w(x) evaluated in x = x0 . Substituting
definition [11], let us assume to have a function f (x), this expression in eq. (14) brings to
and let us define, starting from f (x), the two functions
δ(x − x0 )
Z
F (x, p) = px − f (x) and G(x, p) = ∂F (x, p)/∂x. It is I= dx F (x, p)∆(p, x0 ) 0 , (16)
then possible to define the Legendre transform g(p) of |G (x0 , p)|
f (x) as:
and it appear clear that we can choose ∆(x0 , p) in such
a way that it compensates the extra term in the the de-
g(p) = F (x, p)|G(x,p)=0 , (12) nominator, namely
 
 ∂G(x, p) 
∆(p, x0 ) = 
  = |G0 (x0 , p)|. (17)
where the constraint G(x, p) = 0 is intended to be solved ∂x x=x0
with respect to x. The reader should also keep in mind
that, in order for the Legendre transform to make sense, Note that this definition is fully equivalent and consis-
the initial function f (x) has to be convex, i.e., the func- tent with the one given by Eq. (9). This brings to the
tion must have positive second derivative. It is also worth following result:
to be noted, that the definition of Legendre transform Z
given here is only one of its possible form. Frequently, I= dx F (x, p)δ(x − x0 ) = F (x0 , p). (18)
the Legendre transform is also defined as a constrained
maximization of the function f (x) [11]. For the purposes
Now, since x0 is the value at which the function G(x, p) =
of this paper, however, this definition suites more the
0, the last term in the equality can be rewritten as
needings.
F (x, p)|G(x,p)=0 , thus giving
As an example of calculation of the Legendre trans- Z
form, let us take the simple function f (x) = x2 /2. We
 
I = dx F (x, p)∆(x0 , p)δ G(x, p) dx
then have that F (x, p) = px − x2 /2, G(x, p) = p − x, and
the function G(x, p) vanishes for x0 = p. Substituting = F (x0 , p) = F (x, p)|G(x,p)=0 ≡ g(p) (19)
this into Eq. (12) we have the following:
This is exactly the Legendre transform g(p) of the initial
 function f (x). This result is worth a bit of discussion.
x2 

To start with, let us consider again Eq. (12). There,
g(p) = F (x, p)|G(x,p)=0 = xp −
2  the Legendre transform is implemented by applying the

x=x0 =p
constraint G(x, p) = 0 to the function F (x, p). In that
p2 p2 case, the constraint is applied “ad hoc” in order to pro-
= p2 − = . (13)
2 2 duce the function g(p). In Eq. (16), instead, the same
 4

constraint G(x, p) = 0 is applied to the function F (x, p) respect to the variable q̇, and that q acts simply as a pa-
in a more natural way by means of the Faddeev-Popov rameter in Eq. (23). So said, we can build the analogue
trick, by making the constraint the argument of the Dirac of the function F (x, p) as
delta function. The Legendre transformation is now im-
plemented by selecting (among all the possible values of F (q̇, p; q) = pq̇ − Lho (q, q̇). (24)
x) only those values x0 that make the argument of the
Dirac delta in Eq. (16) vanish. The constraint, in this Notice that we used the semi-colon in the argument of
case, is therefore automatically applied, and the result F (q̇, p; q) to indicate that here q plays the role of a pa-
is the same as in the case of the standard definition of rameter, as the application of the Faddeev-Popov method
Legendre transform as given by Eq. (12). leaves q unchanged. The constraint G(x, p) is now given
We now repeat the example of Sect. III by using the by
Faddeev-popov method given by Eq. (14) to calculate
the Legendre transform of the function f (x) = x2 /2. As ∂F (q̇, p; q) ∂Lho (q, q̇)
before, F (x, p) = xp − x2 /2 and G(x, p) = p − x, that G(q̇, p; q) = = p− = p−mq̇, (25)
∂ q̇ ∂ q̇
vanish for x0 = p. Therefore we have that ∆(p, x0 ) =
|G0 (p, p)| = 1 and
and therefore ∆(q̇0 , p; q) = |G0 (q̇0 , p; q)| = m, where
  q̇0 = p/m is the point at which G(q̇, p; q) vanishes with
δ G(x, p) = δ(x − p). (20) respect to q̇. Notice that G(q̇, p; q) = 0 corresponds to
the usual definition of the canonical momentum in terms
Substituting these results into Eq. (14) then gives: of derivative of the Lagrangian with respect to q̇. This
is the actual constraint that implements the Legendre
p2
Z
I= dx F (x, p)δ(x − p) = F (p, p) = ≡ g(p). (21) transform.
2 In this case, Eq. (14) becomes
This trivial, but instructive, example proves that our Z h i  
definition of Legendre transform via the integral in Eq. I = dq̇ pq̇ − Lho (q, q̇) m δ p − mq̇
(19) is a valid and fully compatible definition of Legendre Z
transform. Moreover, this allows us to give a more intu-
h i  p
= dq̇ pq̇ − Lho (q, q̇) δ q̇ −
itive and clear explanation on how to apply the Faddeev- m
Popov method and what is its physical meaning, by pro- p  p
viding an example of application in a more familiar con- = p − Lho q,
m m
text for the students, than the framework of quantum p2 1 2 2
field theory where this method finds its natural applica- = + mω q ≡ Hho . (26)
2m 2
tion.
As a second, more physical example, let us consider the
problem of calculating the Hamiltonian of an harmonic
oscillator given its Lagrangian V. CONCLUSIONS

1 2 1
Lho (q, q̇) = mq̇ − mω 2 q 2 , (22) In conclusion, we have discussed how the Faddeev-
2 2 Popov method can be used to easily implement the Leg-
where q̇ = dq/dt, m is the oscillator mass and ω is the endre transform of an arbitrary function f (x). As an ex-
characteristic oscillator frequency. As it is well known, ample, we considered the simple case of f (x) = x2 /2 as a
the Hamiltonian is calculated by firstly introducting the direct verification of the validity of our correspondence.
canonically conjugate momentum p = ∂Lho /∂ q̇ = mq̇ We have also presented, as a more physically meaningful
and then by performing a Legendre transform [7]: example, how to use the Faddeev-Popov method to cal-
culate the Hamiltonian of a classical harmonic oscillator
p2 1 starting from the knowledge of its Lagrangian function.
Hho (q, p) = pq̇ − Lho (q, q̇) = + mω 2 q 2 . (23) This provides a novel point of view on the method itself,
2m 2
that only makes use of concepts familiar to the students
We now want to reproduce the same result by applying such as the Legendre transform, without invoking any
the Faddeev-Popov method. First of all, it is worth notic- complicated theoretical framework as quantum field the-
ing that in this case the Legendre transform is made with ory or gauge theory.

[1] J. D. Jackson,Classical Electrodynamics, 3rd edition (Wi- [2] S. Weinberg, The Quantum Theory of Fields, Volume II:
ley, Hoboken, NJ, 1999). Modern Applications (Cambridge University Press, Cam-
 5

bridge, UK, 2005). calculus, Am. J. Phys. 72,753-757 (2004).

[3] A. Das, Field Theory: a Path Integral Approach, 2nd [10] Normally, the Lorentz gauge is considered. However, with
edition (World Scientific Pub. Co., Singapore, 2006). no loss of generality we can assume any gauge condition
[4] L. H. Ryder, Quantum Field Theory, (Cambridge Uni- here.
versity Press, Cambridge, UK, 1985). [11] R. T. Rockafellar, Convex Analysis (Princeton Land-
[5] L. S. Brown, Quantum Field Theory, Revised edition marks in Mathematics and Physics), (Princeton Univer-
(Cambridge University Press, Cambridge, UK, 1994). sity Press, Princeton, NJ, 1970).
[6] M. E. Peskin and D. V. Schroeder, An Introduction to [12] L. M. Brown (editor), Feynman’s Thesis: A New Ap-
Quantum Field Theory, (Westview Press, Boulder, CO, proach to Quantum Theory, (World Scientific Pub Co.,
1995). Singapore, 2005).
[7] H. Goldstein, C. P. Poole and J. L. Safko Classical [13] In this case J µ serves as ancillary variable that highly
Mechanics, 3rd international edition (Addison Wesley, simplifies the calculations. See Ref. [4–6] for a more de-
Boston, MA, 2001). tailed discussion about the role of the current term J µ .
[8] L. D. Landau and E. M. Lifshitz, Statistical Physics: [14] F. W. Byron and R. W. Fuller, Mathematics of Classical
Volume 5 (Course of Theoretical Physics), 3ed edition and Quantum Physics, Revised edition (Dover Pub. Inc.,
(Butterworth-Heinemann, Oxford, UK, 1975). Mineola, NY, 1992).
[9] J. W. Cannon, Connecting thermodynamics to students’