Phase-Space Wick Rotation Explained

• Phase-space Wick rotation is a transformation on the entire phase space that generalizes Wick rotation by complexifying both coordinates and momenta in quantization methods.
• In deformation quantization, this method establishes associative star products on Kähler and pseudo-Kähler manifolds, maintaining Poisson structures but not the *-involution.
• In loop quantum gravity, a complexifier function generates the rotation, mapping Euclidean states to Lorentzian ones while addressing technical issues like singular loci.

Phase-space Wick rotation refers to procedures that implement "Wick rotation" not merely at the level of coordinate time variables but as specific transformations on the entire phase space of classical or quantum systems. This approach underlies both rigorous constructions of deformation quantization on Kähler and pseudo-Kähler manifolds and the canonical relations between Euclidean and Lorentzian formulations in loop quantum gravity (LQG). In these settings, the phase-space Wick rotation is realized either as a holomorphic automorphism of function algebras on reduced spaces or as a canonical transformation generated by a "complexifier" function on infinite-dimensional phase space.

1. Formalism of Wick-type Star Products on Complex Phase Space

A mathematically precise incarnation of the phase-space Wick rotation operates in deformation quantization of Kähler and pseudo-Kähler manifolds (Schmitt et al., 2019). Given integers $n\in\mathbb{N}$, $s\in\{1,\ldots,n+1\}$, consider the phase space $\mathbb{C}^{1+n}$ equipped with coordinates $z^0,\ldots, z^n$ and a quadratic form distinguished by signature variables $v_k$: $$v_k = \begin{cases} +1, & k < s \ -1, & k \geq s \end{cases}$$ The $U(1)$-Hamiltonian action corresponds to the momentum map $J(z) = \sum_{k=0}^n v_k |z^k|^2$. The associated bivector

$$H = \sum_{k=0}^n v_k\,\frac{\partial}{\partial z^k} \otimes \frac{\partial}{\partial \bar{z}^k}$$

determines the "pseudo-Wick" star product, defined for a formal parameter $\lambda$ as

$$f \star g = \sum_{r=0}^\infty \frac{1}{r!} \langle (D_{\text{Sym}}^r f) \otimes (D_{\text{Sym}}^r g), H^{\otimes r} \rangle$$

where $D_{\text{Sym}}$ is the (symmetrized) covariant derivative relevant to the Kähler structure. The first-order term recovers a Poisson bracket of the form

$$\{f, g \} = -2i \sum_{k=0}^n v_k (\partial_{z^k} f\,\partial_{\bar{z}^k} g - \partial_{\bar{z}^k} f\,\partial_{z^k} g)$$

and the product is associative to all orders.

2. Phase Space Reduction and Distinction of Geometries

The construction proceeds by symplectic reduction: restrict to $Z = J^{-1}(1)\subset \mathbb{C}^{1+n}$ and mod out by the $U(1)$ action to obtain the reduced phase space $M_{\rm{red}} = Z/U(1)$. Depending on $s$, $M_{\rm{red}}$ is identified with either the complex projective space $\mathbb{CP}^n$ (Fubini–Study metric, $s=n+1$) or the complex hyperbolic disc $$\mathbb{D}^n = \{ w\in\mathbb{C}^n\,|\,\sum |w^k|^2<1 \}$$ ($s=1$). The Poisson structure, both at the level of bracket and tensors, descends via pullback under the projection and is preserved under the reduction, with explicit, signature-dependent expressions for the reduced bivector $H_{\rm{red}}$.

3. Analytic Completions and the Strict Star Products

There is a convergent subalgebra of polynomial functions $P(\mathbb{C}^{1+n})$, which projects to a $*$-subalgebra $P(M_{\rm{red}})$ in the reduced space. For fixed polynomial $f,g$, the series for the star product $\star_{\rm{red}}|_{\lambda=h}$ is finite except for poles at $h=1/k$, $k\in\mathbb{N}$. One topologizes $P(M_{\rm{red}})$ by embedding in the space of holomorphic functions on a complexification $M_{\rm{red}}^\mathbb{C}$ and completing to an algebra of analytic functions holomorphic on $M_{\rm{red}}^\mathbb{C}$. For values of the deformation parameter $h$ not at the pole set, the product $\star_{\rm{red},h}$ extends to a continuous associative product on $\mathscr{A}(M_{\rm{red}})$, defined as the analytic completion via holomorphic extension [(Schmitt et al., 2019), Theorem 5.26].

4. Signature Change and Wick Rotation as Algebra Isomorphism

The phase-space Wick rotation is realized as a holomorphic map $a^{(s)}$ between reduced spaces $M^{(n+1)}\to M^{(s)}$, explicitly: $$a^{(s)}:\ [ (z, \bar{z}) ] \mapsto [W^{(s)} z, W^{(s)}\bar{z} ]$$ where $W^{(s)}$ is the diagonal matrix shifting signs according to signature. This induces a pullback $\Phi^{(s)}(f) = f\circ a^{(s)}$ on the analytic function algebra. $\Phi^{(s)}$ intertwines the associative algebra operations and Poisson brackets as well as the star products: $$\Phi^{(s)}(f \star_{(n+1),h} g) = \Phi^{(s)}(f)\star_{(s),h}\Phi^{(s)}(g)$$ so at the level of the strict analytic algebras, the Wick rotation implements an algebra isomorphism across different signatures.

5. Failure of $*$-Structure Preservation and Physical Consequences

Despite intertwining all algebraic and Poisson structures, the phase-space Wick rotation $\Phi^{(s)}$ fails to commute with the $*$-involution (pointwise complex conjugation), since $a^{(s)}$ is not anti-holomorphic. This prevents the map from being a $*$-isomorphism. As a result, positivity and reality properties are not preserved:

• For the hyperbolic disc ($s=1$), evaluation at points is a positive linear functional for $h<0$.
• For $\mathbb{CP}^n$ ($s=n+1$), no nontrivial positive functional exists for certain $h$ (e.g., $n=1$). Consequently, the quantum theories obtained after Wick rotation are not unitarily equivalent, and the $*$-structure distinguishing positive states is not mapped across signatures. The associative algebraic identification via Wick rotation does not imply equivalence of quantum mechanical state spaces [(Schmitt et al., 2019), Proposition 6.8, 6.10].

6. Phase-space Wick Rotation in Loop Quantum Gravity

An alternative construction of phase-space Wick rotation arises in loop quantum gravity (Varadarajan, 2018). The original proposal by Thiemann involved canonical transformations, termed "Wick transformations," from the real Ashtekar–Barbero variables suited to Euclidean gravity to the (complex) self-dual Ashtekar variables governing Lorentzian gravity.

The generator of this transformation is the phase-space function ("complexifier")

$T_+ := \frac{\pi}{2G} \int_\Sigma K_a^i E_i^a$

where $K_a^i$ is the extrinsic curvature and $E_i^a$ the densitized triad. A positive, phase-space Wick rotation is then generated by the positive complexifier $C := |T_+|$, a smooth function except on the co-dimension-one set $\{T_+ = 0\}$, where differentiability fails.

The Hamiltonian flow generated by $C$,

$\frac{df}{dt} = \{f, -iC\}$

yields, for $T_+>0$, the self-dual Ashtekar variables and, for $T_+<0$, the anti-self-dual variables. On the singular locus $T_+=0$, the transformation is trivial. Quantum mechanically, $C$ is promoted to a positive, self-adjoint operator $\hat{C}$ on the diffeomorphism-invariant Hilbert space $\mathcal{H}_{\rm diff}$. The Wick rotation is then implemented by the bounded operator $e^{+\hat{C}}$, mapping Euclidean physical states to Lorentzian ones: $\psi_L = e^{+\hat{C}} \psi_E$ with the corresponding Hamiltonian constraints related by conjugation,

$$\hat{H}_L(N) = e^{+\hat{C}}\, \hat{H}_E(N)\, e^{-\hat{C}}$$

This construction ensures a controlled and isometric correspondence between physical state spaces, except on the measure zero set where $T_+=0$ (Varadarajan, 2018).

7. Interpretation, Technical Issues, and Scope

Both deformation quantization and LQG constructions establish phase-space Wick rotation as a transformation acting globally on phase space or its algebra of observables, rather than as a formal substitution of time variables. In deformation quantization, the method applies to strict analytic function algebras and demonstrates the potential for analytic continuation of quantizations across inequivalent signatures. In the context of LQG, the approach provides a mathematically rigorous pathway between Euclidean and Lorentzian sectors, including explicit control over the mapping of physical states and constraints and the preservation (or lack) of positivity structures. However, physical non-equivalence (due to non-preservation of $*$-structure or adjoints) persists, and technical challenges, such as the treatment of singular loci or domain subtleties, remain under active investigation (Schmitt et al., 2019, Varadarajan, 2018).