The boosts and translations do not commute in neither relativistic or nor relativistic systems, please see for example the case of the
Poincare group.
Since $K_i = M_{0i}$ , $(i=1,2,3)$, we get for the Poincare group:
$[K_i, P_j] =i (\eta_{0j}P_i-\eta_{ij}P_0) = -i \delta_{ij}P_0$
Now, The Galilean group can be obtained from the Poincare group by means of Wigner-Inonu contraction, see for example this link.
In the non relativistic limit the rest momentum is dominated by the mass term thus under the contraction the above relation becomes:
$[K_i, P_j] =-i \delta_{ij}M$
The above reasoning shows that the parameter $M$ is the mass.
Considered as an Lie algebra element M commutes with all other generators and cannot be removed by a smooth redefinition of the generators,
this is the reason that it is called a central extension, see for example chapter 3 of
Ballentine's book, where several representations of the Galilean group are constructed.
The consequence is that in each irreducible representation of The Galilean group $M$ must be represented by a scalar (the particle's mass), and two representations with different masses are
unitarily inequivalent.
The formula of the product of a translation and a boost can be directly obtained the cation of the Baker–Campbell–Hausdorff formula.
This formula means that the wavefunctions acquires a phase factor in a transformed inertial frame (in other words, the wavefunction representation is projective). In quantum mechanics a global phase is not measurable, thus no physical consequence is caused
However, Representations corresponding to different masses fall into different superselection sectors, this means that linear superpositions
of wavefunctions of particles with different masses are unphysical, because in this case each component would acquire a different phase which contradicts experiment because differential phases are observable in quantum mechanics.
It is worthwhile to mention that this is exactly the example which was originally analyzed by V. Bargmann in his paper "On unitary ray representations of continuous groups", Ann. of Math.,59 (1954), 1–46.
Update
This update contains the answers to the questions appearing in Aruns comments:
The noncommutativity of the momentum and boost operators is quantum mechanical. The commutators are written in $\hbar= 1$ units, working in general units, the boost-momentum commutator has the form:
$[K_i, P_j] =i \hbar \delta_{ij}M$
The noncommutativity is observable only in the action of the operators on the wave functions. Let me elaborate this point for the case a free particle considered in Ballentine's book.
First, the phase space (i.e., the manifold of initial data) is $\mathbb{R}^6$ covered with the coordinates $\{ \mathbf{q}, \mathbf{v} \}$, which upon quantization satisfy the commutation relations:
$[ q_i, v_j] = \frac{i}{M} \delta_{ij}$
Consider the operators:
The finite translation operator $\mathcal{D}_{q_0} =\exp(i \frac{ M\mathbf{v}. \mathbf{q_0}}{\hbar})$.
The finite boost operator $\mathcal{B}_{v_0} = \exp(i \frac{\mathbf{K}. \mathbf{v_0}}{\hbar}) = \exp(i \frac{M \mathbf{q}. \mathbf{v_0}}{\hbar})$
Consider a wave function on the phase space $\psi(\mathbf{q})$. It is not difficult to see that:
$\mathcal{B}_{v_0} \mathcal{D}_{q_0} \psi(\mathbf{q})= \exp(\frac{i M \mathbf{v_0}.\mathbf{q_0}}{\hbar}) \mathcal{D}_{q_0} \mathcal{B}_{v_0} \psi(\mathbf{q})$
The fourth relativistic momentum components is given by
$ P_0 = \sqrt{M^2 c^4 + p^2 c^2}$
In the non-relativistic limit $c \rightarrow \infty$, the first term dominates and we get $ P_0 \approx M c^2$. Now, you absorb the factor $c^2$ into the definition of the generators, (or work in units c=1). This is essentially
the Wigner-Inonu contraction of the Poincare group to the Galilean group.
I'll give you here the standard argument of the uniatry inequivallence between the quantization of two free particles with different masses $ M_1 \ne M_2$.
Let us use the subscript 1 and 2 for the individual particle operators (Each acting on a different Hilbert space ). If the two representations are unitarily equivalent means that there is an isometry $U$ such that
$\mathbf{K_2} = U \mathbf{K_1}U^{*} $ and $\mathbf{P_2} = U \mathbf{P_1}U^{*} $, thus we obtain $[\mathbf{K_2}, \mathbf{P_2}] = U [\mathbf{K_1}, \mathbf{P_1}]U^{*}$ , which means $M_1 = M_2$, a contradiction.
Unitary inequivalence is similar to representations of $SU(2)$ having different $J$, consquently having different dimensions. In the finite dimensional case, it is obvious that we cannot find an isometry between different dimensional Hilbert spaces.
One might think that all infinite dimensional representations of the Galilean are unitarily equivalent which is not the case.
I wish I could point to a single book that does what you want. The best I can offer is this suggestion: Read (or rather, skim) a few books on axiomatic field theory. Just get a sense for what the physics being described by the axioms is. Draw cartoons. Don't get too hung up on the topology.
I'd start with Haag's book, Local Quantum Physics. Learn about nets of algebras, which are a way of encoding the idea that physical observables actually are located somewhere in spacetime. Also does a nice job of explaining the idea that there's some sort of algebra of observables, and that states -- both pure states and density matrices -- are linear functionals on this algebra. Notice how crucial the fact that spacelike separated observables commute is to matching up the product of operators with the product of observables.
The thing really missing in Haag's book, if you're reading axiomatics as a way of organizing you're thinking about the material in some particle physics book, is a notion of field. The algebras in Haag's book really ought to be generated by 'local observables'. There should be operator-valued functions -- or rather, distributions -- on spacetime, and you should get the algebras in Haag's book by smearing these operators with test functions and then multiplying them. There is a language for this, which is nicely explained in Streater & Wightman's PCT, Spin, Statistics, and All That. This book also explains a critical reconstruction theorem: If you can write down all of the correlation functions of a QFT, then you can recover the Hilbert space & operators directly from these correlation functions.
Thinking of the basic local fields as generators of the algebra of observables is a good idea, but it's more complicated than you might think. The product of two fields at a point is in general singular. Equivalently, many of the correlation functions diverge as you approach the diagonals. You can define non-trivial products by subtracting away these divergences.
It's worth reading Hollands & Wald's paper Axiomatic Quantum Field Theory in Curved Spacetime, at about this point. They axiomatize QFT on globally hyperbolic spacetimes in terms of operator product expansions. Hollands and Wald's point of view is very close to the modern Wilsonian point of view on QFT, which says a) that a QFT is a deformation of a conformal field theory, and b) that a CFT is a collection of local observables obeying an OPE which satisfies the bootstrap condition. You could probably alter their diffeomorphism language to something involving conformal maps and get a definition.
It's also worth spending more time thinking about OPEs and CFTs. Spend some time with diFrancesco et al's Conformal Field Theory or maybe one of the math books on vertex algebras (Kac, Vertex Algebras for Beginners or Frenkel-Ben-Zvi Vertex Algebras & Algebraic Curves)?
Then at last, the path integral. It's not the most general definition of a QFT, but it's very useful. Basically, it gives you a way of writing down the correlation functions, and from the correlation functions, you can read off the OPEs, recover the Hilbert space, etc. The basic idea is best learned first in lattice QFT; I like Montvay & Munster's Quantum Fields on the Lattice. This is also a good time to learn a bit about gauge fields.
The biggest annoyance of the path integral approach is that the funcrtions on field space which are integrable with respect to the continuum measure -- the ones that represent observables in whatever vacuum you're playing in -- can be hard to express in terms of variables that people typically use to construct the measure. This is what renormalization is all about; it gives you a way of using stupid variables and then isolating the main contributions to the correlation functions you're after.
The standard mathematical reference on path integrals is Glimm & Jaffe, Quantum Physics: A Functional Integral Point of View. Most of what you'll want is in Chapter 3,6, or Appendix A. They spend most of the book grinding out a detailed construction of the correlation functions of 2d massive scalar field theory. Figuring out how their constructions are related to standard renormalization language is a worthwhile exercise.
Best Answer
The presumed equivalence between the canonical quantization and the Fock space representation is only a particular case.
The canonical formalism provides only with canonical Poisson brackets. The first step according to Dirac's axioms is to replace the Poisson brackets by commutators and since these commutators satisfy the Jacobi identity, they can be represented by linear operators on a Hilbert space.
Canonical quantization does not specify the Hilbert space.
Finding a Hilbert space where the operators acts linearly and satisfy the commutation relations is a problem in representation theory. This task is referred to as "quantization" in the modern literature.
The problem is that in the case of free fields, this problem does not have a unique solution (up to a unitary transformation in the Hilbert space). This situation is referred to as the existence of inequivalent quantizations or inequivalent representations. The Fock representation is only a special case. Some of the quantizations are called "non-Fock", because the Hilbert space does not have an underlying Fock space structure (i.e., cannot be interpreted as free particles), but there can even be inquivalent Fock representations.
Before, proceeding, let me tell you that inequivalent quantizations may be the areas where "new physics" can emerge because they can correspond to different quantum systems.
Also, let me emphasize, that the situation is completely different in the finite dimensional case. This is because that due to the Stone-von Neumann theorem, any representation of the canonical commutation relations in quantum mechanics is unitarily equivalent to the harmonic oscillator representation. Thus the issue of inquivalent representations of the canonical commutation relations occurs only due to the infinite dimensionality.
For a few examples of inquivalent quantizations of the canonical commutation relations of a scalar field on a Minkowski space-time, please see the following article by: Moschella and Schaeffer. In this article, they construct inequivalent representations by means of Bogoliubov transformation which changes the vacuum and they also present a thermofield representation. In all these representations the canonical operators are represented on a Hilbert space and the canonical commutation relations are satisfied. The Bogoliubov shifted vacuum cases correspond to broken Poincare' symmetries. One can argue that these solutions are unphysical, but the symmetry argument will not be enough in the case of quantization on a general curved nonhomogenous manifold. In this case we will not have a "physical" argument to dismiss some of the inequivalent representations.
The phenomena of inequivalent quantizations can be present even in the case of finite number of degrees of freedom on non-flat phase spaces.
Having said all that, I want nevertheless to provide you a more direct answer to your question (although it will not be unique due to the reasons listed above). As I understand the question, it can be stated that there is an algorithm for passing from the single particle Hilbert space to the Fock space. This algorithm can be summarized by the Fock factorization:
$$ \mathcal{F} = e^{\otimes \mathcal{h}}$$
Where $\mathcal{h}$ is the single particle Hilbert space and $\mathcal{F}$ is the Fock space. As stated before canonical quantization provides us only with the canonical commutaion relations:
$$[a_{\mathbf{k}}, a^{\dagger}_{\mathbf{l}}] = \delta^3(\mathbf{k} - \mathbf{l}) \mathbf{1}$$
At this stage we have only an ($C^{*}$)algebra of operators. The reverse question about the existence of an algorithm starting from the canonical commutation relations and ending with the Fock space (or equivalently, the answer to the question where is the Hilbert space?) is provided by the Gelfand -Naimark-Segal construction (GNS), which provides representations of $C^{*}$ algebras in terms of bounded operators on a Hilbert space.
The GNS construction starts from a state $\omega$ which is a positive linear functional on the algebra $ \mathcal{A}$ (in our case the algebra is the completion of all possible products of any number creation and annihilation operators).
The second step is choosing the whole algebra as an initial linear space $ \mathcal{A}$. In general, there will be null elements satisfying:
$$\omega (A^{\dagger}{A}) = 0$$
The Hilbert space is obtained by identifying elements differing by a null vector:
$$ \mathcal{H} = \mathcal{A} / \mathcal{N} $$
($\mathcal{N} $ is the space of null vectors).
The inner product on this Hilbert space is given by:
$$(A, B) = \omega (A^{\dagger}{B}) $$
It can be proved that the GNS construction is a cyclic representation where the Hilbert space is given by the action of operators on a cyclic "vacuum vector". The GNS construction gives all inequivalent representations of a given $C^{*}$ algebra (by bounded operators). In the case of a free scalar field the choice of a Gaussian state defined by its characteristic function:
$$ \omega_{\mathcal{F} }(e^{\int\frac{d^3k}{E_k} z_{\mathbf{k}}a^{\dagger}_{\mathbf{k}} + \bar{z}_{\mathbf{k}}a^{\mathbf{k}} }) = e^{\int\frac{d^3k}{E_k} \bar{z}_{\mathbf{k}} z_{\mathbf{k}}}$$
Where $z_{\mathbf{k}}$ are indeterminates which can be differentiated by to obtain the result for any product of operators.
The null vectors of this construction will be just combinations vanishing due to the canonical commutation relations (like $a_1 a_2 - a_2 a_1$). Thus this choice has Bose statistics. Also subspaces spanned by a product of a given number of creation operators will be the number subspaces.
The state of this specific construction is denoted by: $\omega_{\mathcal{F}}$, since it produces the usual Fock space. Different state choices may result inequivalent quantizations.