Symplectic Vector Spaces

A symplectic vector space is an even-dimensional vector space equipped with a nondegenerate antisymmetric bilinear form. It is the linear-algebra model of classical phase space near a point.

For a real vector space $V$, a symplectic form is a map

$$\omega:V\times V\to\mathbb R$$

such that

$$\omega(au+bv,w) = a\,\omega(u,w)+b\,\omega(v,w),$$ $$\omega(u,av+bw) = a\,\omega(u,v)+b\,\omega(u,w),$$ $$\omega(u,v)=-\omega(v,u),$$

and nondegeneracy holds:

$$\omega(u,v)=0 \quad \text{for all }v\in V \quad \Longrightarrow \quad u=0.$$

Unlike an inner product, a symplectic form does not measure lengths or angles. It pairs directions in conjugate pairs. In mechanics those pairs are position and momentum directions, and preserving this pairing is what a linear canonical transformation does.

Helpful background. Vector Spaces and Dual Spaces explains bilinear pairings; Phase Space provides the position–momentum example.

The position–momentum pairing

A displacement in position and a displacement in momentum determine an oriented phase-space area. A symplectic form generalizes this pairing to many degrees of freedom. Unlike a metric, it gives no length to a vector; its value on two directions changes sign when they are interchanged.

The Standard Model

The standard symplectic vector space is $\mathbb R^{2n}$ with coordinates

$$z= \begin{pmatrix} q\\ p \end{pmatrix}, \qquad q,p\in\mathbb R^n.$$

For two tangent vectors

$$u= \begin{pmatrix} \delta q\\ \delta p \end{pmatrix}, \qquad v= \begin{pmatrix} \Delta q\\ \Delta p \end{pmatrix},$$

define

$$\omega(u,v) = \sum_{i=1}^n \left( \delta q^i\Delta p_i - \delta p_i\Delta q^i \right).$$

In matrix form,

$$\omega(u,v) = u^T J v,$$

where

$$J= \begin{pmatrix} 0&I\\ -I&0 \end{pmatrix}.$$

Here $I$ is the $n\times n$ identity matrix. The matrix $J$ is antisymmetric,

$$J^T=-J,$$

and invertible,

$$J^{-1}=-J.$$

Antisymmetry says $\omega(u,u)=0$ for every vector $u$. Nondegeneracy says that every nonzero direction has at least one conjugate direction with which it has nonzero symplectic pairing.

Canonical Bases

A basis

$$e_1,\ldots,e_n, f_1,\ldots,f_n$$

is symplectic or canonical when

$$\omega(e_i,e_j)=0, \qquad \omega(f_i,f_j)=0, \qquad \omega(e_i,f_j)=\delta_{ij}.$$

In such a basis the matrix of $\omega$ is the standard matrix $J$ above. The $e_i$ directions are position-like directions and the $f_i$ directions are momentum-like directions.

Every finite-dimensional symplectic vector space admits a canonical basis. This is the linear version of the coordinate statement that, locally, phase space can be written in canonical conjugate pairs. The nonlinear extension becomes the Darboux theorem on symplectic manifolds, but this page only needs the linear statement.

One immediate consequence is that a symplectic vector space has even dimension. There is no nondegenerate antisymmetric bilinear form on an odd-dimensional real vector space. Intuitively, every direction needs a conjugate partner.

Not an Inner Product

It is tempting to read $\omega(u,v)$ as a kind of dot product, but that is wrong.

An inner product is symmetric or Hermitian and positive in the sense that $\langle u,u\rangle>0$ for nonzero $u$. A symplectic form is antisymmetric, so

$$\omega(u,u)=0$$

for every $u$. It does not define a norm.

This difference matters in quantum mechanics. Hilbert space uses an inner product to compute probabilities and adjoints. Classical phase space uses a symplectic form to define Hamiltonian flow and Poisson brackets. Semiclassical methods compare the two structures, but they do not identify them.

Linear Symplectic Maps

A linear map $M:V\to V$ is symplectic if it preserves the symplectic form:

$$\omega(Mu,Mv)=\omega(u,v) \qquad \text{for all }u,v\in V.$$

In the standard coordinates, this becomes

$$M^TJM=J.$$

Such maps form the symplectic group

$$Sp(2n,\mathbb R) = \{M\in GL(2n,\mathbb R):M^TJM=J\}.$$

These are exactly the linear canonical transformations in standard phase-space coordinates. They preserve the canonical Poisson brackets and the Hamiltonian form of the equations.

Taking determinants of $M^TJM=J$ gives

$$(\det M)^2=1.$$

This leaves two algebraic possibilities, but preservation of the nonzero volume form $\omega^n/n!$ fixes $\det M=1$. The converse is false when $n>1$: volume preservation alone is not enough to preserve the symplectic form.

One Degree of Freedom

For $n=1$,

$$J= \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}.$$

Let

$$M= \begin{pmatrix} a&b\\ c&d \end{pmatrix}.$$

Then

$$M^TJM = (\det M)J.$$

Therefore, in one degree of freedom, a real linear map is symplectic exactly when

$$\det M=1.$$

This is why area preservation in the $(q,p)$ plane is enough for one canonical pair. In several degrees of freedom, symplectic preservation is stronger: it preserves each conjugate pairing and the cross-couplings between pairs, not merely total volume.

Basic Examples

A conjugate scaling

$$Q=aq, \qquad P=\frac{p}{a}, \qquad a\ne0,$$

has matrix

$$M= \begin{pmatrix} a&0\\ 0&a^{-1} \end{pmatrix},$$

and is symplectic because $\det M=1$ in one degree of freedom.

A shear

$$Q=q, \qquad P=p+kq$$

has matrix

$$M= \begin{pmatrix} 1&0\\ k&1 \end{pmatrix},$$

and is also symplectic.

For the harmonic oscillator with Hamiltonian

$$H(q,p) = \frac{p^2}{2m} + \frac12m\omega_0^2q^2,$$

the time evolution is linear in $(q,p)$:

$$\begin{pmatrix} q(t)\\ p(t) \end{pmatrix} = \begin{pmatrix} \cos\omega_0 t& \dfrac{\sin\omega_0 t}{m\omega_0} \\ -m\omega_0\sin\omega_0 t& \cos\omega_0 t \end{pmatrix} \begin{pmatrix} q(0)\\ p(0) \end{pmatrix}.$$

This matrix is symplectic. The oscillator flow preserves phase-space area and, more importantly, preserves the canonical pairing between $q$ and $p$.

Relation to Poisson Brackets

In standard coordinates the symplectic form has matrix $J$. To fix the sign, define Hamiltonian vectors by $\omega(X_g,v)=dg(v)$. Since $X_g^T Jv=(\nabla g)^Tv$ for every $v$, we obtain $X_g=J\nabla g$. Therefore

$$\{f,g\} = (\nabla f)^T J \nabla g,$$

when gradients are organized as

$$\nabla f = \begin{pmatrix} \partial f/\partial q\\ \partial f/\partial p \end{pmatrix}.$$

Although the same displayed matrix $J$ appears in both formulas, its matrix inverse is $J^{-1}=-J$. The contraction order in $\omega(X_g,\cdot)=dg$ accounts for this sign. A reliable check is $\{q^i,p_j\}=\delta^i_j$, together with $\dot q^i=\partial H/\partial p_i$ and $\dot p_i=-\partial H/\partial q^i$.

This is why preserving $\omega$ is equivalent to preserving the canonical Poisson brackets. If $M$ is symplectic, then the new linear coordinates have the same bracket relations:

$$\{Q^i,Q^j\}=0, \qquad \{P_i,P_j\}=0, \qquad \{Q^i,P_j\}=\delta^i_j.$$

Relation to the Heisenberg Group

The symplectic pairing also appears in the group law for phase-space translations. In Weyl form, quantum translations by phase-space vectors do not commute exactly. Their commutator phase is controlled by the classical symplectic pairing.

Schematically, if $z$ and $z'$ are phase-space displacement vectors, the central phase contains a factor proportional to

$$\omega(z,z').$$

This is the bridge from classical symplectic geometry to the canonical commutation relations. The detailed group-level construction belongs to Heisenberg Group, while this page supplies the bilinear form being used.

Linearized Hamiltonian Flow

Near a classical trajectory, small deviations obey linear equations. If

$$\dot z=X_H(z),$$

then a variation $\delta z$ satisfies

$$\frac{d}{dt}\delta z = A(t)\delta z,$$

where $A(t)$ is the derivative of the Hamiltonian vector field along the trajectory.

For Hamiltonian systems, the fundamental solution matrix of this variational equation is symplectic. This fact is one reason symplectic matrices appear in semiclassical propagators, Gaussian wave-packet evolution, stability matrices, and Maslov-index calculations.

The finite-dimensional statement here is modest but important: linearized Hamiltonian flow preserves the same symplectic pairing as the original nonlinear flow.

Common Mistakes

• Treating a symplectic form as an inner product.
• Forgetting that symplectic vector spaces are even-dimensional.
• Assuming every determinant-$1$ matrix is symplectic in more than one degree of freedom.
• Thinking canonical coordinates are just a naming convention rather than coordinates adapted to $\omega$.
• Mixing sign conventions for $J$, $\omega$, and the Poisson bracket without checking Hamilton’s equations.
• Saying a quantum unitary transformation is literally the same object as a classical symplectic transformation. The relation is a correspondence, and implementation can involve phases, domains, and double covers.
• Treating preservation of phase-space volume as the whole content of Hamiltonian mechanics.

Exercises

1. Show that $\omega(u,u)=0$ for every vector $u$ if $\omega$ is antisymmetric.

Solution

Antisymmetry gives $\omega(u,u)=-\omega(u,u)$. Therefore $2\omega(u,u)=0$, so over the real numbers $\omega(u,u)=0$.

2. For

$$M= \begin{pmatrix} a&b\\ c&d \end{pmatrix}, \qquad J= \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix},$$

verify that $M^TJM=(ad-bc)J$.

Solution

First compute

$$JM = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix} \begin{pmatrix} a&b\\ c&d \end{pmatrix} = \begin{pmatrix} c&d\\ -a&-b \end{pmatrix}.$$

Then

$$M^TJM = \begin{pmatrix} a&c\\ b&d \end{pmatrix} \begin{pmatrix} c&d\\ -a&-b \end{pmatrix} = \begin{pmatrix} 0&ad-bc\\ bc-ad&0 \end{pmatrix} = (\det M)J.$$

3. Check that the conjugate scaling $Q=aq$, $P=p/a$ preserves the Poisson bracket $\{Q,P\}=1$.

Solution

Using $\{q,p\}=1$ and bilinearity,

$$\{Q,P\} = \left\{ aq, \frac{p}{a} \right\} = \frac{a}{a} \{q,p\} = 1.$$

Thus the scaling is canonical. It stretches one direction and contracts the conjugate direction by the inverse factor.

4. Give an example showing why determinant $1$ is not the same as symplectic when $n>1$.

Solution

In coordinates $(q_1,q_2,p_1,p_2)$, consider

$$M= \begin{pmatrix} 2&0&0&0\\ 0&1/2&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix}.$$

Its determinant is $1$, so it preserves four-dimensional volume. But it sends $q_1$ to $2q_1$ without sending $p_1$ to $p_1/2$. Therefore the bracket of the transformed pair is

$$\{Q_1,P_1\}=2,$$

not $1$. The map is volume-preserving but not symplectic.

5. Explain in words why a symplectic form cannot define a probability norm on Hilbert space.

Solution

A norm must assign a positive size to a nonzero vector. A symplectic form is antisymmetric, so $\omega(u,u)=0$ for every vector $u$. It records conjugate pairing, not length. Quantum probabilities require the Hilbert-space inner product, not the classical phase-space symplectic form.

References

• V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989.
• A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001; author version revised 2006.
• H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison-Wesley, 2002.