Path Integrals for Quadratic Actions

Quadratic actions are exactly solvable because their fluctuations remain Gaussian. A rigorous way to assess the resulting kernel is to construct it as a unitary operator and verify that it implements the intended linear classical motion. This also explains why a divergent-looking prefactor at a caustic can correspond to a perfectly regular unitary evolution. The standard free and oscillator determinant derivations belong to Free-Particle Path Integral and Harmonic-Oscillator Path Integral.

Required background. Oscillatory Integrals fixes Gaussian phases. Schrödinger Representation defines position and momentum on their common Schwartz core.

A quadratic endpoint action generates a linear canonical map

Consider a real symplectic matrix in one degree of freedom,

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

acting by $(x,p_f)^T=M(y,p_i)^T$. The entries have the units required by this map; for example $b$ has units of position divided by momentum.

When $b\ne0$, the quadratic generating function is

$$S_M(x,y)=\frac{dx^2-2xy+ay^2}{2b}.$$

Indeed $p_f=\partial_xS_M$ and $p_i=-\partial_yS_M$ reproduce the classical map. A proposed quantum kernel therefore takes the form

$$K_M(x,y)=\frac{\gamma_M}{\sqrt{2\pi\hbar|b|}} \exp\left(\frac{iS_M(x,y)}{\hbar}\right), \qquad |\gamma_M|=1.$$

The normalization and phase have different roles: the modulus is fixed by unitarity, while the phase requires compatibility with the chosen evolution and composition law.

Fourier factorization proves unitarity

Use the unitary momentum-scaled Fourier transform

$$(\mathcal F_\hbar f)(p)=(2\pi\hbar)^{-1/2} \int e^{-ipy/\hbar}f(y)\,dy.$$

On Schwartz functions, the kernel operator is exactly

$$(U_M\psi)(x)=\gamma_M|b|^{-1/2} e^{idx^2/(2\hbar b)} \mathcal F_\hbar\left(e^{iay^2/(2\hbar b)}\psi(y)\right)(x/b).$$

Both quadratic multipliers have modulus one, the Fourier transform is unitary, and the factor $|b|^{-1/2}$ normalizes the change of variable $p=x/b$. Thus $U_M$ extends uniquely to a unitary operator on all $L^2(\mathbb R)$.

This construction defines the operator even when an undamped multiple Gaussian integral is not absolutely convergent. It also preserves the Schwartz space, where integration by parts verifies

$$U_M^*QU_M=aQ+bP,\qquad U_M^*PU_M=cQ+dP.$$

For example, differentiating the kernel in $y$ gives $U_M(aQ+bP)\psi=xU_M\psi$. The second identity follows similarly. The identities extend to the corresponding self-adjoint closures; they are not unrestricted equalities of differential expressions on arbitrary Hilbert vectors.

The action does not determine the phase branch alone

For free motion, $a=d=1$ and $b=t/m$. The continuous evolution fixes $\gamma_M=e^{-i\pi\operatorname{sgn}(t)/4}$ in this chart. For a general path of symplectic matrices, Gaussian composition produces signature phases when intermediate quadratic forms change sign.

The metaplectic representation organizes these phases: it is a double cover of the symplectic group. A path starting at the identity determines a lift once the quantum generator is the Weyl quantization of the specified homogeneous quadratic Hamiltonian. Adding a scalar to that Hamiltonian would add a further overall dynamical phase while leaving the classical map unchanged.

Thus canonical covariance alone does not fix $\gamma_M$. The initial condition, Schrödinger equation, and group composition must agree. Choosing a fresh principal square root independently at each time can violate those requirements.

A caustic requires another kernel chart

When $b=0$, symplecticity gives $a\ne0$ and $d=1/a$. The classical endpoint relation is $x=ay$, so it cannot be represented by the preceding nonsingular generating function. Up to the phase selected by the same continuous lift, the operator is instead

$$(U_M\psi)(x)=|a|^{-1/2} e^{icx^2/(2\hbar a)}\psi(x/a).$$

Its kernel contains $\delta(y-x/a)$. It is unitary, despite the failure of the $|b|^{-1/2}$ formula. A caustic is a singularity of that coordinate kernel chart, not a divergence of the evolution operator in norm.

For the oscillator, half a classical period gives parity times a phase. The exact distributional limit and its Maslov phase are developed at the canonical oscillator page. The operator formulation explains why that continuation is compulsory rather than a repair to an otherwise divergent physical evolution.

Exercise: normalization is forced by the operator norm

Replace the prefactor of $e^{iS_M/\hbar}$ by an arbitrary constant $C$, with $b\ne0$. Find the condition on $|C|$ for the resulting operator to be unitary.

Solution

The same Fourier factorization gives $\|U\psi\|^2=2\pi\hbar|b||C|^2\|\psi\|^2$. Hence $|C|=(2\pi\hbar|b|)^{-1/2}$. The condition leaves a unit-modulus phase undetermined. This isolates precisely what a norm check can certify and what still requires the composition and generator conditions.

References

• Folland, Gerald B., Harmonic Analysis in Phase Space, Annals of Mathematics Studies 122, Princeton University Press, 1989, Chapter 4 — the metaplectic representation and quadratic integral operators.
• Hall, Brian C., Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — canonical transformations and the oscillator realization.