Phase space and the tautological form

Let’s consider a physical system described by its configuration space $\mathcal{Q}$. We saw in the previous lecture that the evolution of the system on $\mathcal{Q}$ can be described by finding the extremum of its action functional, expressed as the time integral of the Lagrangian function $L(q,\dot{q})$ defined on the tangent space $T\mathcal{Q}$, on which velocities are defined. As a reminder, a point of $T\mathcal{Q}$ is a doublet $(q,v)$, where $q$ is a generalized position and $v$ a velocity/tangent vector.

Alternatively, it is possible to describe its evolution on the cotangent space of $\mathcal{Q}$, also known as the phase space $\mathcal{P} = T^{*} \mathcal{Q}$. A point in phase space is given by the doublet of a point of $\mathcal{Q}$ and a covector, noted $p$, that we will identify with momentum. Hence $x=(q,p)\in \mathcal{P}$.

Working in phase space instead of tangent space has several advantages. The most important one being that it renders apparent a special geometrical structure of classical mechanics: symplectic geometry. This geometry allows to investigate general quantization procedures, looking for a general and rigorous path to go from classical to quantum systems. This is for example the case of the so-called geometric quantization.

Before going through this lecture, we recommend to first have a look at our previous lectures on Hamiltonian mechanics and Phase space for statistical physics.

A general one form field on $\mathcal{Q}$ would be written $ P_i(q)\text{d}q^i \in \Gamma(T^{*}\mathcal{Q})$ (here and everywhere below, if an index is repeated, it is summed over). When thinking of $\mathcal{P}=T^{*}\mathcal{Q}$ as its own independent space, we consider every possible values of $P_i(q)$ at each point $q$, and promote it to its own coordinate, that we name $p_i$. More precisely: it is always possible to find local coordinates on a chart $U \subseteq \mathcal{Q}$ such that a point $q\in \mathcal{Q}$ is labeled by coordinates $q^i$. Once the $q_i$ are chosen, we label $p_i$ the associated one-form coordinate such that $q_i$ and $p_i$ form canonical coordinates. As such, a general point $x\in \mathcal{P}$ is given by the tuple $x=(q^1, …, q^{3\mathcal{N}},p_1, …, p_{3\mathcal{N}})$. Where $\mathcal{N}$ is the number of particles if $\Pi$ describes a set of point-like particles allowed to move in three dimensions. A general one form field on $\mathcal{P}$ is written

\[\eta = A_i(q,p)\text{d}q^i + B^i(q,p)\text{d}p_i\]

which is a section of $T^{*}(\mathcal{P}) = T^{*}(T^{*}\mathcal{Q})$, $\eta \in \Gamma(T^{*}(\mathcal{P}))$ and the coordinates $A_i$ and $B^i$ are functions of $C^{\infty}(\mathcal{P})$, (i.e. $A_i: \mathcal{P}\to\mathbb{R}$ and $B^i: \mathcal{P}\to\mathbb{R}$). Similarly, a general vector field on $\mathcal{P}$ is written as

\[Y= u^i(q,p) \partial_{q^i} + v_i(q,p)\partial_{p_i}\]

with $Y\in \Gamma(T\mathcal{P})$ and the coordinates $u^i$, $v_i$ are functions of $C^{\infty}(\mathcal{P})$.

Symplectic geometry

The phase space inherits a peculiar geometry through the existence of the so-called tautological one-form. It is also known as Liouville one-form, the Poincaré one-form, the canonical one-form, or the symplectic potential. Once a coordinate chart is chosen, the tautological one form is defined as the special choice of one form field with $A_i(q,p)=p_i$ and $B_i=0$, that is:

\[\boxed{\theta = p_i \text{d} q^i}\]

As discussed below the value of $\theta$ is independent of the choice of local coordinates, and admits a coordinate independent definition.

More details on the tautological form

As a fiber bundle, $\mathcal{P}$ is equipped with a projection map $\pi$ to its base space:

\[\pi:T^*\mathcal Q\to\mathcal Q\]

For a point $(q,p)\in T^*\mathcal Q$, we consider a tangent vector of phase space at this point:

\[X\in T_{(q,p)}(T^*\mathcal Q),\]

Using the pushforward \(\pi_*:T(T^*\mathcal Q)\to T\mathcal Q\), it is possible to obtain a vector $\pi_*(X)$ on $T_q\mathcal{Q}$ from the vector $X \in T\mathcal{P}$. Once pushed forward on the configuration space, the one-form $p:T\mathcal{Q}\to \mathbb{R}$ – coresponding to the original phase-space coordinate where $X$ was defined – can act on it to give a number as \(p(\pi_*(X))\). As such, the tautological one-form is defined by

\[\boxed{ \theta_{(q,p)}(X) = p\bigl(\pi_*X\bigr) }\]

Or, equivalently:

\[\boxed{\theta_{(q,p)} = p \circ \pi_*}\]

Computing this value at every point $x=(q,p)$ of $\mathcal{P}$, one obtain a 1-form field.

Clearly, this definition does not depend on any choice of coordinatization of $\mathcal{P}$. Choosing local coordinates immediately gives

\[\theta=p_i\,\mathrm dq^i.\]

Indeed, consider a general vector at the point $q,p$:

\[X= u^i \partial_{q^i} + v_i\partial_{p_i}\]

The pushforward is:

\[\pi_*(X) = u^i\partial_{q^i}\]

Now, the one-form $p_i\text{d}q^i \in T^*_q(\mathcal{Q})$ can act on it to give

\[p(\pi_*(X)) = p_i u^i\]

As subtle point: while $p=p_i\text{d}q^i$ is a one form on $\mathcal{Q}$, \(p \circ \pi_*= p_i \text{d}q^i + 0 \text{d}p_i\) is a one form on $\mathcal{P}$. While both take the same form in coordinates, they are thus different objects.

A symplectic manifold is a pair $(M,\omega)$, with $M$ a smooth manifold and $\omega$ a symplectic form. A 2-form field $\omega \in \Omega^2(\mathcal{P}_s)$ is said to be a sympletic form if

$\theta$, already present necessarily on $\mathcal{P}$, endows the phase space with a symplectic structure. Indeed, taking its exterior derivative, one obtains the symplectic form defined as

\[\boxed{\Omega := -\text{d}\theta}\]

The minus sign is inserted here only by convention, in order to recover expressions more familiar to physicists (several textbook use instead the opposite convention $\Omega=\text{d}\theta$ and one should be careful about it, as it will change the sign of multiple expressions below). $\Omega$ can be expressed in local coordinates as

\[\boxed{\Omega=\text{d}q^i\wedge \text{d}p_i}\]

The pair $(\mathcal{P},\Omega)$ is a symplectic manifold.

Proof that $\Omega$ is a symplectic form

On the definition of $\Omega$:

First, we recall that, by definition of $\text{d}$, for $p$-form $\alpha$ and a $k$ form $\beta$:

\[\text{d}(\alpha \wedge \beta) = \text{d}\alpha \wedge \beta + (-1)^k \alpha \wedge \text{d}\beta\]

Hence

\[\begin{align} \text{d}\theta &= \text{d}( p_i \text{d}q^i)\\ &= \text{d}( p_i \wedge \text{d}q^i)\\ &= \text{d}p_i\wedge \text{d}q^i - p_i \wedge \text{d}^2q^i\\ &= \text{d}p_i\wedge \text{d}q^i \end{align}\]

Because $\text{d}^2=0$. Then

\[\begin{align} \Omega &= -\text{d}\theta\\ &= - \text{d}p_i\wedge \text{d}q^i \\ &= \text{d}q^i\wedge \text{d}p_i \end{align}\]

From the antisymmetry of the wedge product ($\alpha \wedge \beta = -\beta \wedge \alpha$).

$\Omega$ is closed:

Now, $\Omega$ is clearly closed by being the exterior derivative of a form (of the form $-\theta$) and from the defining property $\text{d}^2=0$.

$\Omega$ is non-degenerate:

Let $X=A_i \partial_{q^i} + B^i \partial_{p_i}$.

Hence $\Omega$ is non-degenerate.

As a two-form, $\Omega$ defines a product between two vectors $X,Y \in T\mathcal{P}$ as $\Omega(X,Y)$. In coordinates, if $X=A^i\partial_{q^i}+ B_i\partial_{p_i}$ and $Y= u^i\partial_{q^i} + v_i\partial_{p_i}$, we have

\[\boxed{\Omega(X,Y)= A^i v_i - B_iu^i.}\]

From this equation, we see that, when applied to a unique vector, we obtain $\Omega(X,X)=0$. Hence $\Omega$ plays an analogous role as the metric $g$ on Riemannian space, endowing it with a very different geometric structure (which is anti-symmetric instead of symmetric and gives a zero “length” to vectors. Compare for example to the simplest Euclidian metric which would give instead $g(X,Y)=A^iu^i + B_iv_i$).

Proof \[\begin{align} \Omega(X,Y) &= (\text{d}q^i\wedge \text{d}p_i)(X,Y) \nonumber \\ &= (\text{d} q^i \otimes \text{d} p_i -\text{d} p_i \otimes \text{d} q^i)(X_F,Y) \nonumber \\ &= ( \text{d} q^i \otimes \text{d} p_i)(X,Y) - ( \text{d} p_i \otimes \text{d} q^i)(X,Y)\nonumber \\ & = \text{d} q^i(X) \text{d} p_i(Y) - \text{d} p_i(X) \text{d} q^i(Y) \nonumber \\ &= A^i v_i - B_iu^i \end{align}\]

Recalling that, from the definition of the exterior derivative, $\text{d}f(v)=v(f)$, and thus, for example:

\[\begin{align} \text{d}q^i(X) = X(q^i) &= (A^i\partial_{q^i}+ B_i \partial_{p_i})(q^i) \\ &= A^i \frac{\partial q^i}{\partial q^i} + B_i \frac{\partial q^i}{\partial p_i}\\ &= A_i \end{align}\]

If $F$ is a smooth function (observable) defined on $\mathcal{P}$, we can define the Hamiltonian vector field $X_F$ of $F$ as

\[\begin{equation} \boxed{ \text{d} F = \Omega(X_F,\cdot) } \end{equation}\]

which is sometimes written equivalently using the interior derivative as $\text{d} F = i_{X_F} \Omega$.

In coordinates, the vector solving this equation takes the form:

\[\begin{equation} \boxed{ X_F = \frac{\partial F}{\partial p_i}\partial_{q^i}- \frac{\partial F}{\partial q^i}\partial_{p_i}} \end{equation}\]
Proof

Let $X_F = (A^i \partial_{q^i} + B_i \partial_{p_i})$. We want to find $A^i$ and $B_i$ such that $\text{d} F = i_{X_F} \Omega$. Let $Y= (u^i \partial_{q^i} + v_i \partial_{p_i})$ be an arbitrary vector. We have

\[\begin{align} i_{X_F} \Omega(Y) &= \Omega(X_F,Y) = \text{d} F(Y) \end{align}\]

From the symplectic product formula derived previously, we find that the left hand side gives:

\[\begin{align} \Omega(X_F,Y) = A^iv_i - B_i u^i \end{align}\]

The right hand side gives:

\[\begin{align} \text{d} F(Y)& = \left(\frac{\partial F}{\partial q^i} \text{d} q^i + \frac{\partial F}{\partial p_i} \text{d} p_i\right)(u^i \partial_{q^i} + v_i \partial_{p_i}) \nonumber \\ &= \frac{\partial F}{\partial q^i} u^i + \frac{\partial F}{\partial p_i} v_i \end{align}\]

Equating the two sides, we find that $A^i = \frac{\partial F}{\partial p_i}$ and $B_i = -\frac{\partial F}{\partial q^i}$ and hence

\[\begin{equation} X_F = \frac{\partial F}{\partial p_i}\partial_{q^i}- \frac{\partial F}{\partial q^i}\partial_{p_i} \end{equation}\]

Poisson Bracket and Hamiltonian vector fields

As discussed before, physical observables are functions on the phase space manifold $\mathcal{P}$, noted $\mathcal{O}= C^{\infty}(\mathcal{P})$. Let $F,G \in C^{\infty}(\mathcal{P})$ be two observables, we introduce the Poisson bracket product on them as

\[\begin{equation} \boxed{ \lbrace F,G\rbrace := \Omega(X_F,X_G)} \end{equation}\]

The Poisson bracket satisfies many interesting properties. First, it is clearly an anti-symmetric operation, that is:

which follows directly from the anti-symmetry of $\Omega$.

Proof

Furthermore, the existence of the Poisson bracket equips the observable space $\mathcal{O}$ with a Lie algebra structure as it satisfies:

Proof

The Poisson bracket also obeys Leibniz rule for the product of functions, making it a sort of differential operator:

Proof

The Poisson bracket can be rewritten in several equivalent ways, highlighting some of these properties. For instance, using the definition of $X_F$ we also have

\[\begin{equation} \lbrace F,G\rbrace = \text{d} G(X_F) = X_F(G) = \mathcal{L}_{X_F}G \end{equation}\]
Proof of the different expressions

First recall again, that a vector $v$ is an object that “eats” a function to give a number $v(f)$ and that the exterior derivative of $\text{d}f$ is a one form, which “eats” a vector to give a number $\text{d}f(v)$ such that:

\[\text{d}f(v) = v(f)\]

The Lie derivative applied on a function has the exact same expression (they differ when applied on other objects):

\[\mathcal{L}_vf = v(f)\]

So it will be straightforward to prove the last expression involving it.

Now, we recall that, by definition of $X_F$:

\[\text{d}F = \Omega(X_F,.)\]

Hence, it is easy to see that:

\[\text{d}F(X_G) = \Omega(X_F,X_G)\]

which is the definition of $\lbrace F, G \rbrace $ and prooves our first expression.

Furthermore, from the definition of $\text{d}$ above, we immediately get the second expression:

\[\text{d}F(X_G) =X_G(F)\]

We also note that, from the anti-symmetry of $\Omega$ discussed previously, we also have the two other expressions:

\[\lbrace F,G \rbrace = - X_F(G) = - \text{d}G(X_F)\]

In local coordinates, the Poisson bracket takes the form:

\[\begin{equation} \boxed{ \lbrace F,G\rbrace = \frac{\partial F}{\partial q^i}\frac{\partial G}{\partial p_i} - \frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q^i} } \end{equation}\]
Proof

We saw previously that

\[X_F = \frac{\partial F}{\partial p_i}\partial_{q^i}- \frac{\partial F}{\partial q^i}\partial_{p_i}\]

and similarly

\[X_G = \frac{\partial G}{\partial p_i}\partial_{q^i}- \frac{\partial G}{\partial q^i}\partial_{p_i}\]

Hence, using again the formula for the symplectic product:

\[\begin{align} \lbrace F, G\rbrace &= \Omega(X_F,X_G)\\ &= -\frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q^i} + \frac{\partial F}{\partial q^i}\frac{\partial G}{\partial p_i} \\ &= \frac{\partial F}{\partial q^i}\frac{\partial G}{\partial p_i} - \frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q^i} \end{align}\]

Let’s now consider the Lie algebra formed by the observables. Let’s consider a generator $G$ which generates transformations of any observable $f$ under a variation of another observable $F$ as $\delta f=e^{GF}f$. If the observable $G$ is a conserved quantity under variations of the observable $F$, then, one has, for any observable $f$:

\[\begin{equation} \boxed{ X_G(f) = \lbrace f,G\rbrace = \frac{\partial f }{\partial F} } \end{equation}\]

We say that $G$ is the generator of the evolution of the system under variations of $F$.

Proof

Under a variation of $F$, any observable $f$ transforms as \(\begin{equation} \delta f = \frac{\partial f}{\partial q_i} \delta q_i + \frac{\partial f}{\partial p_i}\delta p_i \end{equation}\)

where $\delta q_i$ and $\delta p_i$ are the changes of position in phase space due to a change of $F\to F+\delta F$. Let’s now assume that this change is generated by $G$ as $f=fe^{G F}$ such that $\delta f = G \delta F$.

Now if $G$ is conserved

\[\begin{equation} \delta G = \frac{\partial G}{\partial q_i}\frac{\partial q_i}{\partial F} + \frac{\partial G}{\partial p_i}\frac{\partial p_i}{\partial F} =0 \end{equation}\]

such that \(\begin{equation} \frac{\partial G}{\partial q_i}\frac{\partial q_i}{\partial F} = -\frac{\partial G}{\partial p_i}\frac{\partial p_i}{\partial F} \end{equation}\)

which is satisfied for

\[\begin{equation} \frac{\partial G}{\partial p_i} = \frac{\partial q_i}{\partial F} \qquad \frac{\partial G}{\partial q_i} = -\frac{\partial p_i}{\partial F} \end{equation}\]

approximating the derivatives as $\partial G \partial X \simeq \delta G/\delta X$, we find

\[\begin{equation} \delta q_i = \frac{\partial G}{\partial p_i}\delta F \qquad \delta p_i = -\frac{\partial G}{\partial q_i}\delta F \end{equation}\]

Re-inserting these expressions in our first expression for $\delta f$, we find that

\[\begin{equation} \delta f = \frac{\partial f}{\partial q_i}\frac{\partial G}{\partial p_i}\delta F - \frac{\partial f}{\partial p_i}\frac{\partial G}{\partial q_i}\delta F = \lbrace f,G\rbrace \delta F \end{equation}\]

such that

\[\begin{equation} \frac{\partial f}{\partial F}= \lbrace f,G\rbrace \end{equation}\]

$X_G$ is then the gradient of $F$. Integral curves and vector flows

\[\begin{equation} \frac{\partial \phi^G}{\partial F} = X_G \end{equation}\]

As such, we have $\lbrace F,G\rbrace = \partial_F F = 1$. $F$ and $G$ are said to be canonical variables.

The generator is conserved under variations of $F$ as $\partial_FG=\lbrace G,G\rbrace = 0$.

Any observable $f$ such that $\lbrace f,G\rbrace =0$ is called a conserved quantity under variations of $F$, signifying that $f$ remains unchanged through the $F$ evolution of the system generated by $G$ ($\partial_F f=0$).

Time evolution

The Hamiltonian function $H$ is defined as the observable generating time translations

\[\begin{equation} X_H(f) := \lbrace f, H\rbrace = \frac{\partial f}{\partial t} \end{equation}\]

$H$ is the generator of the time $t$ evolution/translations and $\lbrace t,H\rbrace =1$.

This gives back Hamilton’s equations of motion for $f=q_i$ and $f=p_i$:

\[\begin{equation} \frac{\partial q_i}{\partial t}= \frac{\partial H}{\partial p_i}\qquad \frac{\partial p_i}{\partial t}= -\frac{\partial H}{\partial q_i} \end{equation}\]

The symplectic product between two vectors is preserved by time evolution.

Other observables

Position and momentum satisfy

\[\begin{equation} \lbrace q^i,p_j\rbrace =\delta^i_{j} \end{equation}\]

such that they are canonical variables for $i=j$. We find that

\[\begin{equation} X_{p_i}=\lbrace f,p_i\rbrace = \frac{\partial f}{\partial q^i} \end{equation}\]

$p_j$ is thus the generator of the space $q^j$ translations. And the other way around

\[\begin{equation} X_{q_i}(f)=\lbrace f,q_i\rbrace = \frac{\partial f}{\partial p^i} \end{equation}\]

such that $q_i$ can be understood to generate space translations.

Similarly $L_i$ is the generator of the spatial rotations $\theta$ around the axis $i$.

\[\begin{equation} \lbrace f,L_i\rbrace = \frac{\partial f}{\partial \theta} \end{equation}\]

Similar considerations can be done for spin (quantum mechanics), boosts (special relativity) and gauge charge (gauge theories) later.

Of course, this discussion echoes strongly with our discussion of the symmetries in quantum mechanics. As such, it is clear that the continuous symmetries present on the Hilbert space are deeply related with the symmetries on the classical phase space, where the commutator $[.,.]$ playing the role of the Poisson bracket $\lbrace .,.\rbrace$. Finding the exact connection between both is the whole topic of quantization procedures which will be the topic of later discussions.

Phase space as a vector space

\[\Omega_{ij} = \begin{pmatrix} &0 &\mathbb{I}_n\\ &-\mathbb{I}_n &0 \end{pmatrix}\] \[\lbrace\Omega(y_1,.),\Omega(y_2,.)\rbrace = \Omega(y_1,y_2)\]

Going from tangent to contangent space

The action:

\[S = \theta(X_H)\]