Dirac Brackets and Time-Dependent Constraints

Abstract

We provide a simultaneous derivation of the Dirac bracket and of the equations of motion for second-class constrained systems when the constraints are time-dependent. The necessity of time-dependent gauge-fixing conditions is shown in the example of parameterized mechanics and is illustrated geometrically.

1. Introduction

Constrained systems play a major role in modern physics, most notably in gauge theories and in general relativity. Dirac established a theory for the canonical quantization of constrained systems [1], introducing a new bracket, known today as a Dirac bracket, which satisfies all of the desirable properties for a Poisson bracket (linearity in both arguments, Leibnitz rule, antisymmetry, and the Jacobi identity), and which preserves the form of Hamilton’s equations of motion when written in terms of brackets.

In the case of pure gauge theories, gauge-fixing conditions which are time-independent can be set, but in the case of diffeomorphism-invariant theories, such as general relativity, the gauge-fixing conditions have to be time-dependent.

In Dirac’s original work, the Hamiltonian and the constraints are time-independent. The straightforward extension of Dirac’s method to the case of time-dependent constraints was developed in Ref. [2]. Meanwhile, other approaches to this problem have been developed, such as the extension of phase space to include time as a canonical variable [3,4,5], the modern differential geometric approaches of Refs. [6,7], and further applications [8].

Here, we provide an alternative derivation of the expression for Dirac brackets, and one which concomitantly provides an alternative route to solve the problem of time-dependent constraints provided in Ref. [2]. We also geometrically illustrate the need for time-dependent gauge-fixing conditions in theories which are invariant under time reparameterizations.

2. Gauge-Fixed Constrained Systems

In this section, we will not review Dirac’s method, which may already be familiar to many readers, and which is very clearly outlined in Dirac’s original work [1] and in many other works [3,9,10,11,12,13]; rather, we will start from a Hamiltonian which is already in the form

$$H=H_0+\sum _{a=1}^{2M}{u}_a{c}_a,$$

(1)

arising from Dirac’s procedure, and where to each first-class constraint a “gauge-fixing” constraint has been added by hand, with the whole set of constraints becoming second-class.

The antisymmetric matrix

$$M_{ab}=\left\{c_a,c_b\right\}$$

(2)

is, by definition of second-class property, non-singular,

$$detM\ne 0.$$

(3)

Ideally, for singular systems, one would like to be able to perform a canonical transformation to a set of phase space variables containing as a subset canonical pairs which are unconstrained and provide a representation of reduced phase space [11]; that is, which form a set of physical variables that capture the physical sector of the theory [12,14] with unambiguous dynamics. However, not only may such a transformation not exist globally, but also, in general, one will not be able to isolate the variables of reduced phase space. Therefore, one may resortto gauge-fixing; that is, to add as many extra constraints as there are first-class constraints, such that the whole set of constraints becomes second-class. This procedure may still be of limited validity due to Gribov obstructions [9], but this is what we assume has been performed in Equation (1), as we are specifically interested in the study of time-dependent constraints.

Dirac showed that requiring that time evolution preserves the second-class constraints one can solve for the $u_a$ in Equation (1) and set the second-class constraints strongly to zero, and that the time evolution of a phase space variable is then given by

$$\dot{F}={\left\{F,H_0\right\}}_D+{\displaystyle \frac{\partial F}{\partial t}},$$

(4)

where (summation over repeated indices is understood)

$${\left\{F,G\right\}}_D=\left\{F,G\right\}-\left\{F,c_a\right\}{M}_{ab}^{-1}\left\{c_b,G\right\}$$

(5)

is the Dirac bracket.

The straightforward extension of Dirac’s method to time-dependent constraints was performed in Ref. [2], where it was shown that Equation (4) should be replaced by

$$\dot{F}={\left\{F,H_0\right\}}_D+{\left({\displaystyle \frac{\partial F}{\partial t}}\right)}_D,$$

(6)

with

$${\left({\displaystyle \frac{\partial F}{\partial t}}\right)}_D={\displaystyle \frac{\partial F}{\partial t}}-\left\{F,c_a\right\}{M}_{ab}^{-1}{\displaystyle \frac{\partial c_b}{\partial t}}.$$

(7)

3. Parameterized Mechanics

We consider the example of a theory describable by an action principle, from an action

$$S={\int }_{t_1}^{t_2}\mathcal{L}\left(x^{\mu },{\dot{x}}^{\mu }\right)dt$$

(8)

which is invariant under redefinitions of the evolution parameter,

$$t\to f\left(t\right).$$

(9)

This is a diffeomorphism-invariant theory in one dimension. We use the calligraphic letter $\mathcal{L}$ to denote the Lagrangian while utilizing the capital letter L to denote another function, for reasons which will become clear below.

The Lagrangian $\mathcal{L}$ cannot depend explicitly on t and it must be homogeneous of the first kind of the derivatives of the velocities [1,11],

$$\mathcal{L}\left(x^{\mu },\lambda {\dot{x}}^{\mu }\right)=\lambda \mathcal{L}\left(x^{\mu },{\dot{x}}^{\mu }\right),$$

(10)

with $\mu =0,\dots ,d$. Moreover, if we assume that one of the configuration space variables, $x^0$, is a monotonous function of time, $d{x}^0/dt>0$, this condition can be stated as

$$\mathcal{L}\left(x^{\mu },{\dot{x}}^{\mu }\right)=L\left(x^0,x^i,{\displaystyle \frac{{\dot{x}}^i}{{\dot{x}}^0}}\right){\dot{x}}^0,$$

(11)

with $i=1,\dots ,d$. The momenta conjugate to the configuration space variables are

$$\begin{array}{ccc}\hfill p^0& =& L-{\displaystyle \frac{\partial L}{\partial \left({\dot{x}}^i/{\dot{x}}^0\right)}}{\displaystyle \frac{{\dot{x}}^i}{{\dot{x}}^0}}\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill p^i& =& {\displaystyle \frac{\partial L}{\partial \left({\dot{x}}^i/{\dot{x}}^0\right)}},\hfill \end{array}$$

(13)

and the Hamiltonian vanishes,

$$\mathcal{H}=p^0{\dot{x}}^0+p^i{\dot{x}}^i-\mathcal{L}=0.$$

(14)

Assuming that the relation (13) between the $p^i$ and the ${\dot{x}}^i/{\dot{x}}^0$ is invertible (if it is not, then further constraints show up—but we assume it is, in order to concentrate on the reparameterization invariance alone), one can write Equation (12) in the form

$$p^0+H\left(x^0,x^i,p^i\right)=0,$$

(15)

with

$$H={\displaystyle \frac{\partial L}{\partial \left({\dot{x}}^i/{\dot{x}}^0\right)}}{\displaystyle \frac{{\dot{x}}^i}{{\dot{x}}^0}}-L,$$

(16)

which is clearly recognizable as the Legendre transform of L in the variables ${\dot{x}}^i/{\dot{x}}^0$; that is, the Hamiltonian associated with the function L, seen as a Lagrangian. This is why we reserved the calligraphic letters $\mathcal{L}$ and $\mathcal{H}$ for the original Lagrangian and Hamiltonian of the theory.

Equation (15) is a constraint, and because we assumed (13) to be invertible, it is the only constraint, and hence a first-order one. Therefore, the time evolution is undetermined, which we should have expected because we have the freedom to choose any parameterization that we want for the evolution variable t.

Now, we shall resort to gauge-fixing, as indicated in Section 2. Adding the constraint $x^0=0$ seems to be a good choice because it forms a second-class pair with the constraint (15): $\left\{x^0,p^0+H\left(x^0,x^i,p^i\right)\right\}=1$. However, the Hamiltonian vanishes strongly, $\mathcal{H}=0$, and using Equation (4), one would obtain $\dot{F}\left(x^{\mu },p^{\mu }\right)=0$ for any function in phase space; there would be no dynamics! Of course, $x^0=0$ is incompatible with our assumption that $d{x}^0/dt>0$, but the same problem would arise for any constraint of the form $f(x^0,p^0,x^i,p^i)=0$, such that $\left\{f,p^0+H\right\}\ne 0$.

The point is that the transformation (9) that leaves the action (8) invariant is not a pure gauge transformation in the sense that the transformation is not only among the phase space variables; rather, it involves the evolution parameter, while for pure gauge transformations, gauge-fixing conditions that involve only phase space variables can be used. In the case of the invariance under reparameterizations of the evolution variable, the “gauge-fixing” conditions must necessarily involve the evolution variable itself, since that is the variable that must be fixed.

This is illustrated in Figure 1. On the left hand side of the figure is an example where the variable $x^0$ is pure gauge ($x^1$ is some other variable, added for illustrative purposes—it may be looked upon as the rest of the phase space); that is, the system is invariant under $x^0\to x^0+\alpha$, and on the right hand side of the figure, there is an example where the system is invariant under reparameterizations $t\to f\left(t\right)$. In both cases, evolution is undetermined, with dotted lines representing equivalent solutions. In the first case, the gauge can be fixed with $x^0=0$. Then, the solution is unique (solid line). In the second case, setting $x^0=0$ (first solid line) freezes the dynamics ($x^1=const.$), but setting $x^0=t$ preserves the dynamics (second solid line).

The examples in Figure 1 also illustrate why Equation (4) must be improved when the constraints are time-dependent. After eliminating the variable $x^0$, one has

$${\left({\displaystyle \frac{\partial F}{\partial t}}\right)}_{x^1=cont}={\displaystyle \frac{\partial F}{\partial x^0}}{\displaystyle \frac{d{x}^0}{dt}}+{\displaystyle \frac{\partial F}{\partial t}}.$$

(17)

Setting $x^0=0$, we obtain

$${\left({\displaystyle \frac{\partial F}{\partial t}}\right)}_{x^1=cont}={\displaystyle \frac{\partial F}{\partial t}}.$$

(18)

However, setting $x^0=t$ we obtain

$${\left({\displaystyle \frac{\partial F}{\partial t}}\right)}_{x^1=cont}={\displaystyle \frac{\partial F}{\partial x^0}}+{\displaystyle \frac{\partial F}{\partial t}}.$$

(19)

It is therefore the term $\partial F/\partial t$ that must be corrected in Equation (4).

Returning to our theory, adding instead the gauge-fixing constraint

$$C\left(x^0,x^i,p^i\right)-t=0$$

(20)

to (12), with

$$\left\{C,p^0+H\right\}\ne 0,$$

(21)

one obtains, using (6) and (7) instead of (4),

$$\dot{F}={\displaystyle \frac{\partial F}{\partial t}}+{\displaystyle \frac{\left\{F,p^0+H\right\}}{\left\{C,p^0+H\right\}}}.$$

(22)

As for the quantization of the theory, since $\mathcal{H}=0$, there is no immediate Schrödinger picture, but the system can be described in the Heisenberg picture because one can compute $\dot{F}$ using Equation (22).

Furthermore, after some manipulation, one arrives, starting from Equation (22), at

$$\dot{F}={\left\{F,H\right\}}_D+{\displaystyle \frac{\partial F}{\partial t}}+\left\{F,p^0\right\}+{\displaystyle \frac{\left\{C,F\right\}\left\{p^0,H\right\}}{\left\{C,p^0+H\right\}}}.$$

(23)

Hence, a Schrödinger picture can be recovered with the Hamiltonian given by H once one solves the constraints; the last three terms in this equation can be interpreted as the explicit variation with respect to time. This is clear if one chooses $C=x^0$ and solves the constraints for $x^0$ and $p^0$. Then, Equation (23) reads as follows:

$$\dot{F}={\left\{F,H\right\}}_D+{\displaystyle \frac{\partial F}{\partial t}}+{\displaystyle \frac{\partial F}{\partial x^0}}-{\displaystyle \frac{\partial F}{\partial p^0}}{\displaystyle \frac{\partial H}{\partial x^0}}.$$

(24)

The issue of recoverability of a Schrödinger picture in the general case of time-dependent constraints is discussed in Refs. [15,16,17].

The action (8) can be obtained by the inverse procedure, starting from

$$S={\int }_{x_{0\left(1\right)}}^{x_{0\left(2\right)}}L\left(x^0,x^i,{\displaystyle \frac{d{x}^i}{d{x}^0}}\right)d{x}^0$$

(25)

and parameterizing the variable $x^0=x^0\left(t\right)$ [1,11]. That is why it is called parameterized mechanics.

4. General Relativity

General relativity is a field theory, and as such, its treatment is much more elaborate than that of mechanics, which has a discrete number of degrees of freedom. We will not review the extension of analytical mechanics to field theories, which is available elsewhere in many textbooks [9,10,11,13], nor the several complications that may arise when imposing gauge-fixing conditions, such as the existence or not of global gauge-fixing conditions, the presence of surface terms, etc., (for a review of many of these matters, see [18] or [19]). The sole purpose of this section is to provide an important example where time-dependent constraints may show up, as the imposition of time-dependent gauge-fixing conditions is one of several attempts at dealing with the Hamiltonian formulation and general relativity.

The action for general relativity can be written in the form

$$S=\int dt\int d^{3}x\left({\displaystyle \frac{1}{2}}{\pi }^{ij}{\dot{g}}_{ij}-NH-N^i{P}_i\right),$$

(26)

where H and $P_i$ are functions of the space–space components of the metric $g_{ij}$ and their conjugate momenta ${\pi }^{ij}$, and N and $N^i$ are functions that involve the space–time and time–time components of the metric. Since the time derivatives of these latest components of the metric do not show up in the action, the function N and the vector $N^i$, called the lapse and the shift, respectively, act as Lagrange multipliers for the constraints H and $P_i$, called the Hamiltonian constraint and the momentum constraint, respectively. This is the so-called ADM decomposition of the action [18].

Therefore, general relativity is a totally constrained theory, with its Hamiltonian density being a combination of four constraints. These four constraints arise from the invariance under the four-dimensional (per spacetime point) group of diffeomorphisms in four dimensions, and they turn out to be first-class. Since the space–space part of the metric has six independent components, one arrives at the well-known counting of $6-4=2$ degrees of freedom per point for general relativity. One way to fix the gauge is to add four extra constraints,

$$C^{\alpha }\left(x\right)=X^{\alpha }\left({\pi }^{ij}\left(x\right),g_{ij}\left(x\right)\right)-x^{\alpha }=0$$

(27)

to the existing H and $P_i$, such that, together, the eight of them form a set of second-class constraints. Here, $X^{\alpha }$ are some of the functions of the canonical variables at the point x. This can be interpreted as choosing a coordinate system [1,18,19,20], and it is the analogue of Equation (20). Indeed, the original constraints of the theory arise precisely because of the freedom to choose the coordinate system, just like the constraint (15) appears in parameterized mechanics because of the freedom to choose the evolution parameter.

5. Analytical Mechanics in Matrix Form

We shall use the notation in [21], which we find more practical for the present purpose. In this notation, the set of canonical coordinates $x_i$ and $p_i$ is composed into a column vector, and the same is performed for the derivatives of a function F with respect to the canonical variables,

$$\eta =\left[\begin{array}{c}{x}_1\\ \dots \\ x_N\\ p_1\\ \dots \\ p_N\end{array}\right]{\displaystyle \frac{\partial F}{\partial \eta }}=\left[\begin{array}{c}\partial F/\partial x_1\\ \dots \\ \partial F/\partial x_N\\ \partial F/\partial p_1\\ \dots \\ \partial F/\partial p_N\end{array}\right].$$

(28)

With the help of the matrix

$$\mathbf{J}=\left[\begin{array}{cc}0& 1_{N\times N}\\ -1_{N\times N}& 0\end{array}\right],$$

(29)

Hamilton’s equations of motion can be succinctly written as

$$\dot{\eta }=\mathbf{J}{\displaystyle \frac{\partial H}{\partial \eta }},$$

(30)

with H being the Hamiltonian. The Poisson bracket between two functions F and G becomes

$$\left\{F,G\right\}={\left({\displaystyle \frac{\partial F}{\partial \eta }}\right)}^T\mathbf{J}{\displaystyle \frac{\partial G}{\partial \eta }}$$

(31)

and the time evolution of an arbitrary function F is

$$\dot{F}=\left\{F,H\right\}+{\displaystyle \frac{\partial F}{\partial t}}.$$

(32)

6. Time-Dependent Constraints

Let us now admit that the Hamiltonian contains $2M$ second-class constraints $c_a$, which we also write in the form of a column vector

$$c\left(\eta ,t\right)=0.$$

(33)

Its time derivative is

$$\dot{c}=\mathbf{C}\dot{\eta }+{\displaystyle \frac{\partial c}{\partial t}}=0.$$

(34)

It must vanish, as the constraints are to be maintained with evolution. Here, we have defined the $2M\times 2N$ matrix

$$C_{ai}={\displaystyle \frac{\partial c_a}{\partial {\eta }_i}}.$$

(35)

Equation (34) is a simple linear equation. It admits a solution if, and only if [22]

$$\left(\mathbf{1}-{\mathbf{CC}}^{-1}\right){\displaystyle \frac{\partial c}{\partial t}}=0,$$

(36)

where ${\mathbf{C}}^{-1}$ is a pseudoinverse of $\mathbf{C}$; that is, a $2N\times 2M$ matrix satisfying

$${\mathbf{CC}}^{-1}\mathbf{C}=\mathbf{C}.$$

(37)

Its solution is [22]

$$\dot{\eta }=\left(\mathbf{1}-{\mathbf{C}}^{-1}\mathbf{C}\right)Z-{\mathbf{C}}^{-1}{\displaystyle \frac{\partial c}{\partial t}},$$

(38)

for some Z.

Demanding that Equation (38) is applicable to all cases, even in the absence of constraints, when c = 0 and $\mathbf{C}=0$, one must have

$$Z=\mathbf{J}{\displaystyle \frac{\partial H}{\partial \eta }},$$

(39)

since we know that in the absence of constraints, Equation (30) holds. Hence,

$$\dot{\eta }=\left(\mathbf{1}-{\mathbf{C}}^{-1}\mathbf{C}\right)\mathbf{J}{\displaystyle \frac{\partial H}{\partial \eta }}-{\mathbf{C}}^{-1}{\displaystyle \frac{\partial c}{\partial t}}.$$

(40)

Now, we can compute the total derivative of a function $F\left(\eta ,t\right)$

$$\dot{F} \begin{array}{cc}=& {\left({\displaystyle \frac{\partial F}{\partial \eta }}\right)}^T\dot{\eta }+{\displaystyle \frac{\partial F}{\partial t}}=\hfill \\ =& {\left({\displaystyle \frac{\partial F}{\partial \eta }}\right)}^T\left(\mathbf{1}-{\mathbf{C}}^{-1}\mathbf{C}\right)\mathbf{J}{\displaystyle \frac{\partial H}{\partial \eta }}-{\left({\displaystyle \frac{\partial F}{\partial \eta }}\right)}^T{\mathbf{C}}^{-1}{\displaystyle \frac{\partial c}{\partial t}}+{\displaystyle \frac{\partial F}{\partial t}}=\hfill \\ =& {\left({\displaystyle \frac{\partial F}{\partial \eta }}\right)}^T\mathbf{J}{\displaystyle \frac{\partial H}{\partial \eta }}-{\left({\displaystyle \frac{\partial F}{\partial \eta }}\right)}^T{\mathbf{C}}^{-1}\mathbf{CJ}{\displaystyle \frac{\partial H}{\partial \eta }}+{\displaystyle \frac{\partial F}{\partial t}}-{\left({\displaystyle \frac{\partial F}{\partial \eta }}\right)}^T{\mathbf{C}}^{-1}{\displaystyle \frac{\partial c}{\partial t}}\hfill \end{array}$$

(41)

We want to rewrite this equation in the form

$$\dot{F}={\left\{F,H\right\}}_D+{\left({\displaystyle \frac{\partial F}{\partial t}}\right)}_D$$

(42)

with a new “Poisson bracket” ${\left\{F,H\right\}}_D$ that depends linearly on F and H, and a new “partial derivative with respect to time” ${\left(\partial F/\partial t\right)}_D$ which is linear in F. Then, we must have

$$\begin{array}{ccc}\hfill {\left\{F,G\right\}}_D& =& \left\{F,G\right\}-{\left({\displaystyle \frac{\partial F}{\partial \eta }}\right)}^T{\mathbf{C}}^{-1}\mathbf{CJ}{\displaystyle \frac{\partial G}{\partial \eta }}\hfill \end{array}$$

(43)

$$\begin{array}{ccc}\hfill {\left({\displaystyle \frac{\partial F}{\partial t}}\right)}_D& =& {\displaystyle \frac{\partial F}{\partial t}}-{\left({\displaystyle \frac{\partial F}{\partial \eta }}\right)}^T{\mathbf{C}}^{-1}{\displaystyle \frac{\partial c}{\partial t}}.\hfill \end{array}$$

(44)

We have not yet determined the pseudoinverse ${\mathbf{C}}^{-1}$ (recall that the pseudoinverse is not unique), and neither have we showed that Equation (36) holds. Because $\mathbf{C}$ has a maximal rank (otherwise the constraints would not be independent), ${\mathbf{CC}}^T$ is invertible and ${\mathbf{C}}^T{\left({\mathbf{CC}}^T\right)}^{-1}$ is a pseudoinverse (in fact, a right pseudoinverse, and the Moore–Penrose pseudoinverse). This shows that Equation (36) holds. However, using this pseudoinverse would make the Dirac bracket fail the requirement of antisymmetry in the exchange of F with G. As we shall see, imposing this further requirement singles out one pseudoinverse.

The first parcel on the right hand side of Equation (43) is certainly antisymmetric because it is the Poisson bracket. It is therefore enough to require antisymmetry for the second parcel,

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{\partial F}{\partial \eta }}\right)}^T{\mathbf{C}}^{-1}\mathbf{CJ}{\displaystyle \frac{\partial G}{\partial \eta }}=-{\left({\displaystyle \frac{\partial G}{\partial \eta }}\right)}^T{\mathbf{C}}^{-1}\mathbf{CJ}{\displaystyle \frac{\partial F}{\partial \eta }}=\\ \hfill =-{\left[{\left({\displaystyle \frac{\partial G}{\partial \eta }}\right)}^T{\mathbf{C}}^{-1}\mathbf{CJ}{\displaystyle \frac{\partial F}{\partial \eta }}\right]}^T=-{\left({\displaystyle \frac{\partial F}{\partial \eta }}\right)}^T{\mathbf{J}}^T{\left({\mathbf{C}}^{-1}\mathbf{C}\right)}^T{\displaystyle \frac{\partial G}{\partial \eta }}.\end{array}$$

(45)

This can only be satisfied for all F and G if (notice that ${\mathbf{J}}^T=-\mathbf{J}$)

$$\begin{array}{c}\hfill {\mathbf{C}}^{-1}\mathbf{CJ}=\mathbf{J}{\left({\mathbf{C}}^{-1}\mathbf{C}\right)}^T\Rightarrow \\ \hfill \Rightarrow {\mathbf{C}}^{-1}{\mathbf{CJC}}^T={\mathbf{JC}}^T{\left({\mathbf{C}}^{-1}\right)}^T{\mathbf{C}}^T=\mathbf{J}{\left({\mathbf{CC}}^{-1}\mathbf{C}\right)}^T={\mathbf{JC}}^T.\end{array}$$

(46)

Since we know that

$$\mathbf{M}={\mathbf{CJC}}^T={\displaystyle \frac{\partial c_a}{\partial {\eta }_i}}{J}^{ij}{\displaystyle \frac{\partial c_b}{\partial {\eta }_j}}=\sum _{i=1}^N{\displaystyle \frac{\partial c_a}{\partial x_i}}{\displaystyle \frac{\partial c_b}{\partial p_i}}-{\displaystyle \frac{\partial c_a}{\partial p_i}}{\displaystyle \frac{\partial c_b}{\partial x_i}}$$

(47)

is invertible [1], we finally obtain

$${\mathbf{C}}^{-1}={\mathbf{JC}}^T{\mathbf{M}}^{-1}.$$

(48)

Substitution into Equations (43) and (44) yields

$$\begin{array}{ccc}\hfill {\left\{F,G\right\}}_D& =& \left\{F,G\right\}-{\left({\displaystyle \frac{\partial F}{\partial \eta }}\right)}^T{\mathbf{JC}}^T{\mathbf{M}}^{-1}\mathbf{CJ}{\displaystyle \frac{\partial G}{\partial \eta }}\hfill \end{array}$$

(49)

$$\begin{array}{ccc}\hfill {\left({\displaystyle \frac{\partial F}{\partial t}}\right)}_D& =& {\displaystyle \frac{\partial F}{\partial t}}-{\left({\displaystyle \frac{\partial F}{\partial \eta }}\right)}^T{\mathbf{JC}}^T{\mathbf{M}}^{-1}{\displaystyle \frac{\partial c}{\partial t}}.\hfill \end{array}$$

(50)

Or, back to indices notation,

$${\left\{F,G\right\}}_D\begin{array}{cc}=& \left\{F,G\right\}-{\displaystyle \frac{\partial F}{\partial {\eta }_i}}{J}_{ij}{\displaystyle \frac{\partial c_a}{\partial {\eta }_j}}{\left[{\displaystyle \frac{\partial c_a}{\partial {\eta }_m}}{J}_{mn}{\displaystyle \frac{\partial c_b}{\partial {\eta }_n}}\right]}^{-1}{\displaystyle \frac{\partial c_b}{\partial {\eta }_k}}{J}_{kl}{\displaystyle \frac{\partial G}{\partial {\eta }_l}}=\hfill \\ =& \left\{F,H\right\}-\left\{F,c_a\right\}{\left\{c_a,c_b\right\}}^{-1}\left\{c_b,G\right\}\hfill \end{array}$$

(51)

$${\left({\displaystyle \frac{\partial F}{\partial t}}\right)}_D\begin{array}{cc}=& {\displaystyle \frac{\partial F}{\partial t}}-{\displaystyle \frac{\partial F}{\partial {\eta }_i}}{J}_{ij}{\displaystyle \frac{\partial c_a}{\partial {\eta }_j}}{\left[{\displaystyle \frac{\partial c_a}{\partial {\eta }_m}}{J}_{mn}{\displaystyle \frac{\partial c_b}{\partial {\eta }_n}}\right]}^{-1}{\displaystyle \frac{\partial c_b}{\partial t}}=\hfill \\ =& {\displaystyle \frac{\partial F}{\partial t}}-\left\{F,c_a\right\}{\left\{c_a,c_b\right\}}^{-1}{\displaystyle \frac{\partial c_b}{\partial t}}.\hfill \end{array}$$

(52)

To summarize our results, we showed that one can work on the constraint surface, defined by $c_a=0$, provided that one replaces the Poisson bracket with the Dirac bracket

$${\left\{F,G\right\}}_D=\left\{F,G\right\}-\left\{F,c_a\right\}{M}_{ab}^{-1}\left\{c_b,G\right\},$$

(53)

with $M_{ab}=\left\{c_a,c_b\right\}$, and the partial derivative with respect to time with

$${\left({\displaystyle \frac{\partial F}{\partial t}}\right)}_D={\displaystyle \frac{\partial F}{\partial t}}-\left\{F,c_a\right\}{M}_{ab}^{-1}{\displaystyle \frac{\partial c_b}{\partial t}}.$$

(54)

Therefore, we provided an alternative and simultaneous derivation of Equations (5) and (7).

7. Discussion

Time-dependent gauge-fixing conditions are important in a number of situations; namely, in general relativity. The straightforward method of dealing with them in conjunction with the Dirac bracket that we studied in this paper is one possible way of tackling the problem, for which we provided a new and simultaneous derivation of the relevant results. Furthermore, it is helpful to understand the reasons why, and in which situations, one has to work with time-dependent constraints when they arise from gauge-fixing, for which we provided a simple pedagogical and graphical example.

The reader should not get the wrong impression that this derivation overlooked dynamical aspects. In fact, it arises from the same principles used in standard textbooks, namely that the Hamiltonian H generates the equations of motion, Equation (38) together with Equation (39), and the requirement of time preservation of the constraints, expressed in Equation (34).