Deformation quantization as an appropriate guide to ontic structure

Abstract

Karim Thébault has argued that for ontic structural realism to be a viable ontology it should accommodate two principles: physico-mathematical structures it deploys must be firstly consistent and secondly substantial. He then contends that in geometric quantization, a transitional machinery from classical to quantum mechanics, the two principles are followed, showing that it is a guide to ontic structure. In this article, I will argue that geometric quantization violates the consistency principle. To compensate for this shortcoming, the deformation quantization procedure will be offered.

Appendices

Appendix A: Symplectic Geometry

In the following, I will use standard notations (Tu 2010; Lee 2013) to denote differential geometric notions. In particular, I will denote the set of smooth functions on Manifold M by $C^{\infty }(M)$, the tangent space to M at p by $T_pM$, the space of smooth k-vector fields on M by ${\mathfrak {X}}^k(M)$ and the space of smooth k-forms on M by $\varOmega ^k(M)$. My exposition of symplectic geometry formulation of CM is essentially extracted from Da Silva (2001), Lee (2013, Ch. 22) and Esposito (2014, Ch. 2).

Symplectic manifold Let M be a smooth manifold. Structure $(M,\omega )$ is symplectic manifold if:

• (a) M is a smooth manifold;

• (b) $\omega :{\mathfrak {X}}^1(M)\times {\mathfrak {X}}^1(M)\rightarrow C^{\infty }(M)$ is a 2-form ($\omega \in \varOmega ^2(M)$) such that:

  • (b.1) $d\omega =0$, where d is the exterior derivative ($\omega$ is closed);

  • (b.2) $\forall p \in M, V_p \in T_p M: \omega _p(W_p,V_p)=0\rightarrow W_P=0$ ($\omega$ is non-degenrate).

  $\omega$ is called the symplectic 2-form.

Hamiltonian vector field Let $(M,\omega )$ be a symplectic manifold and H, the Hamiltonian function, a smooth function on M ($H\in C^{\infty }(M)$). $X_H \in {\mathfrak {X}}^1(M)$ is called the Hamiltonian vector field associated with H if $\omega (X_H,.)=dH$.

Poisson structure (first characterization) Let $(M,\omega )$ be a symplectic manifold. $(C^{\infty }(M),\circ , \left\{ ,\right\} )$ is called the Poisson Structure if:

• (a) $\circ : C^{\infty }(M)\times C^{\infty }(M)\rightarrow C^{\infty }(M)$ is a communicative multiplication;

• (b) $\left\{ ,\right\} : C^{\infty }(M)\times C^{\infty }(M)\rightarrow C^{\infty }(M)$ is defined as $(f,g)\mapsto \left\{ f,g\right\} :=\omega (X_f,X_g)$, where $X_f$ and $X_g$ are the Hamiltonian vector fields associated with the smooth functions f and g.

Poisson structure (second characterization) Let $(M,\omega )$ be a symplectic manifold. $(C^{\infty }(M),\circ , \alpha )$ is called the Poisson Structure if:

• (a) $\circ : C^{\infty }(M)\times C^{\infty }(M)\rightarrow C^{\infty }(M)$ is a communicative multiplication;

• (b) $\alpha : \varOmega ^1(M)\times \varOmega ^1(M) \rightarrow C^{\infty }(M)$ is a bi-vector field ($\alpha \in {\mathfrak {X}}^2(M))$ such that $\left[ \alpha ,\alpha \right] _S=0$, which $\left[ ,\right] _S: {\mathfrak {X}}^k(M)\times {\mathfrak {X}}^l(M)\rightarrow {\mathfrak {X}}^{k+l-1}(M)$ is the Schouten-Nijenhuis bracket defined as:

  $$\begin{aligned} \begin{aligned} \left[ \alpha ,\alpha \right] _S= \alpha ^{ij}(\partial _i\alpha ^{kl})\partial _j\wedge \partial _k\wedge \partial _l \end{aligned} \end{aligned}$$

  (A.1)

where $\partial ^{ij}$s are bi-vector coefficients in local coordinates. $\alpha$ is called the Poisson bi-vector field and relates to the symplectic 2-form $\omega$ by:

$$\begin{aligned} \begin{aligned} \alpha (df,dg)=\omega (X_f,X_g) \end{aligned} \end{aligned}$$

(A.2)

Hamiltonian system Strcuture $(M_H, \omega _H, H)$ is called the Hamiltonian system if:

• (a) $(M_H,\omega _H)$ is a symplectic manifold such that:

  • (a.1) $M_H={\mathbb {R}}^{2n}$ with coordinate maps $q_1,\ldots ,q_n,p_1,\ldots ,p_n$;

  • (a.2) $\omega _H= \sum \limits _{j=1}^{n} dq^{i}\wedge dp^{j}$;

• (b) $H=C^{\infty }(M)$ is the Hamiltonian function.

It can be shown that (Da Silva 2001, p. 107):

$$\begin{aligned} \begin{aligned} X_H= \sum _{j=1}^{n} \left( \frac{\partial H}{\partial p_j}\frac{\partial }{\partial q_j}-\frac{\partial H}{\partial p_j}\frac{\partial }{\partial q_j}\right) \end{aligned} \end{aligned}$$

(A.3)

which satisfies the Hamiltonian equations. The Darboux theorem (Lee 2013, p. 571) guarantees that for every 2n-dimensional symplectic manifold $(M,\omega )$ and for every $p\in M$, we can find a smooth chart $(U, q_1,\ldots ,q_n,p_1,\ldots ,p_n)$ centered at p in which $\omega = \sum \nolimits _{j=1}^{n} dq^{i}\wedge dp^{j}$. The collection of these smooth charts forms an atlas for $(M,\omega )$.

Structure of classical mechanics (SCM) Let $(M,\omega , H)$ be a symplectic manifold equipped with the Hamiltonian function. The classical mechanics is presented by the associated Poisson structure, i.e. $(C^{\infty }(M),\circ , \alpha )$.

Appendix B: Formal Concepts

Deformation quantization proceeds on the basis of the framework of formal power series. In the following, its key notions and structures will be defined. For more details, see Gracia-Saz (2003), Laurent-Gengoux et al. (2012) and Esposito (2014).

Formal power series with coefficients in an algebra Let A be an algebra over the commutative ring K. Formal Power series with coefficient in A is defined as:

$$\begin{aligned} \begin{aligned} a=\sum _{n=0}^{\infty } t^na_n; \quad a_n\in A \end{aligned} \end{aligned}$$

(B.1)

where t is a formal indeterminate. We denote the set of formal power series by A[t].

Formal power series with coefficients in a ring Let K be a commutative ring. Formal Power series with coefficient in K is defined as:

$$\begin{aligned} \begin{aligned} b=\sum _{n=0}^{\infty } t^n b_n; \quad b_n\in K \end{aligned} \end{aligned}$$

(B.2)

We denote the set of formal power series by K[t].

Formal ring Let K[t] be a commutative ring. $K[\![ t ]\!]=(K[t], +, \circ )$, the formal ring, has the ring structure in which addition is defined by:

$$\begin{aligned} \begin{aligned}&+: K[t], \times K[t] \rightarrow K[t]\\&(a,b)\mapsto a+b=\sum _{n=0}^{\infty } t^n (a_n+b_n); a_n,b_n \in K \end{aligned} \end{aligned}$$

(B.3)

and multiplication by:

$$\begin{aligned} \begin{aligned}&\circ : K[t] \times K[t] \rightarrow K[t]\\&(a,b)\mapsto a\circ b=\sum _{n=0}^{\infty } t^n (c_n); \; c_n=\sum _{k=0}^{\infty } a_n \circ b_{n-k}; \; a_n,b_n \in K \end{aligned} \end{aligned}$$

(B.4)

Formal algebra Let A[t] be an algebra. $A[\![ t ]\!]=(A[t], \circ )$, the formal algebra, is an algebra in which multiplication is defined by:

$$\begin{aligned} \begin{aligned}&\circ : A[t] \times A[t] \rightarrow A[t]\\&(a,b)\mapsto a\circ b=\sum _{n=0}^{\infty } t^n (c_n); \; c_n=\sum _{k=0}^{\infty } a_n \circ b_{n-k}; \; a_n,b_n \in A \end{aligned} \end{aligned}$$

(B.5)

Regarding these definitions, the algebra of smooth functions on manifold M is denoted by $C^\infty (M)[\![ t ]\!]=(C^\infty (M)[t], \circ )$. To define the formal structure of classical mechanics as a formal Poisson structure, we need first to define the formal k-vector field on M.

Formal K-vector field Let $X_n$’s be k-vector fields on M ($X_n \in {\mathfrak {X}}^k)$. The formal k-vector field X as a formal power series is defined as:

$$\begin{aligned} \begin{aligned} X=\sum _{n=0}^{\infty } t^n X_n \end{aligned} \end{aligned}$$

(B.6)

The space of formal k-vector fields is denoted by ${\mathfrak {X}}^k[\![ t ]\!]$. Given an SCM $(C^{\infty }(M),\circ , \alpha )$, the formal Poisson bi-vector field is:

$$\begin{aligned} \begin{aligned} \alpha _t=\left( \sum _{n=0}^{\infty } t^n \alpha _n\right) \in {\mathfrak {X}}^2[\![ t ]\!] \end{aligned} \end{aligned}$$

(B.7)

where $\alpha _i$’s are skew-symmetric vector fields such that $[\alpha _t,\alpha _t]_S=0$.

Formal structure of classical mechanics (FSCM) Let $(C^{\infty }(M),\circ , \alpha )$ be an SCM. $(C^\infty (M)[t], \circ , \alpha _t )$ is an FSCM, if:

• a) $C^\infty (M)[\![ t ]\!]=(C^\infty (M)[t], \circ )$ is the formal algebra associated with the algebra $(C^\infty (M), \circ )$;

• b) $\alpha _t$ is the formal Poisson bi-vector field by setting $\alpha _0=\alpha$.

Equivalent FSCMs $(C^\infty (M)[t], \circ , \alpha _t )$ and $(C^\infty (M)[t], \circ , {\tilde{\alpha }}_t )$ are equivalent if there exists a formal vector field $X=\sum _{n=0}^{\infty } t^n X_n$ such that:

$$\begin{aligned} \begin{aligned} {\tilde{\alpha }}_t=exp(t{\mathcal {L}}_X)\alpha _t \end{aligned} \end{aligned}$$

(B.8)

where ${\mathcal {L}}_X$ is the Lie derivative. The associated equivalence class is denoted by $[A_t]$.

Formal star product Let $A[\![ t ]\!]$ be a formal algebra over the formal ring $K[\![ t ]\!]$. The formal star product is a $K[\![ t ]\!]$-bilinear map:

$$\begin{aligned} \begin{aligned}&\star : A[t] \times A[t] \rightarrow A[t]\\&(f,g)\mapsto f \star g = \sum _{n=0} ^ {\infty } B_n(f,g)t^n; \; f,g\in A \end{aligned} \end{aligned}$$

(B.9)

where $B_n$’s$:A \times A \rightarrow A$ are bi-differential operators, and $f\star g = f \circ g$ for $t=0$. The star product extends to A[t] by:

$$\begin{aligned} \begin{aligned} \left( \sum _{k=0} ^ {\infty } f_k t^k\right) \star \left( \sum _{l=0} ^ {\infty } g_l t^l\right) = \sum _{n=0} ^ {\infty } \left( \sum _{k+l+m=n} B_m(f_k,g_l)t^n\right) ; \; f_k,g_l\in A \end{aligned} \end{aligned}$$

(B.10)

Now we can deform FSCM by the formal star product.

Deformed formal structure of classical mechanics (DFSCM) Suppose that a classical system is re-presented by an FSCM denoted by $(C^\infty (M)[t], \circ , \alpha _t )$. The associated quantized system is re-presented by the deformed structure denoted by $(C^\infty (M)[t], \circ , \star _{\alpha _t})$.

Equivalent DFSCMs $(C^\infty (M)[t], \circ , \star _{\alpha _t} )$ and $(C^\infty (M)[t], \circ , \star _{{\tilde{\alpha }}_t})$ are equivalent if there exists a gauge action:

$$\begin{aligned} \begin{aligned}&D: A[t]\rightarrow A[t]\\&f\mapsto D(f) =D_0(f)+\sum _{n=1} ^ {\infty } D_n(f)t^n \end{aligned} \end{aligned}$$

(B.11)

(where $D_i$s are differential operators and $D_0(f)=f$) such that:

$$\begin{aligned} \begin{aligned} f \star _{{\tilde{\alpha }}_t} g= D(E(f)\star _{\alpha _t} E(g)) \end{aligned} \end{aligned}$$

(B.12)

where E is the inverse of D in the group of gauge actions. The associated equivalence class is denoted by $[\star _{A_t}]$.

Kontsevich’s Theorem (existence and classification of quantization) Let $(C^{\infty }(M),\circ , \alpha )$ be the SCM. There exists a one-to-one correspondence between the equivalence classes of FSCMs on it, $\left\{ [A_t]\right\}$, and the equivalence classes of DFSCMs, $\left\{ [\star _{A_t}]\right\}$.