Why is it so that even in the succeding theory of physics to Newtonians mechanics, the conceptual scaffolding of the theory still transfers?

Question

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1. Restating the Question in Plain Language

A student is wondering why the basic ideas that we learn in Newtonian mechanics – things like position, momentum, energy, force, mass, time, etc. – keep showing up, often with the same names, when we move to more modern theories such as quantum mechanics (QM) and general relativity (GR).

The question can be broken down into three parts

Why do these concepts survive the transition to newer theories?
What is the “core” of Newtonian mechanics that is still useful in the other theories?
Why is it justified to treat that core as the underlying conceptual backbone of physics?

Below we will answer each part step‑by‑step, showing the logical chain that connects the old and the new theories.

2. Detailed Explanation (Step‑by‑Step)

Step 1 – Identify the “conceptual scaffolding’’ of Newtonian mechanics

Newtonian mechanics rests on a small set of primitive concepts and mathematical structures:

Primitive concept	Symbolic/Mathematical representation	Physical role
Space (where things are)	(\mathbf{x}\in\mathbb{R}^3) (a point vector)	Provides a stage for the system
Time (when things happen)	(t\in\mathbb{R})	Orders events, gives dynamics
Mass (inertia)	(m) (scalar, positive)	Relates force to acceleration
Force (cause of change)	(\mathbf{F})	Determines how (\mathbf{x}(t)) changes
Momentum	(\mathbf{p}=m\mathbf{v})	Conserved quantity for isolated systems
Energy	(E = T+V = \frac12 m v^2 + V(\mathbf{x}))	Conserved quantity (if system is time‑invariant)
Newton’s second law	(\mathbf{F}=m\mathbf{a})	Equation of motion

These concepts are tied together by two mathematical ideas:

Differential calculus (rates of change, acceleration).
Symmetry → conservation laws (Noether’s theorem in later language).

Step 2 – Recognise the universal aspects of those concepts

Even though Newton’s specific law ((\mathbf{F}=m\mathbf{a})) is replaced, the structures above are universal:

Universal aspect	Reason it survives
Configuration space (the set of all possible positions)	Any physical theory must describe what a system can be. In QM it becomes a Hilbert space; in GR it becomes a curved manifold.
Time parameter (ordering of events)	Dynamics needs a notion of “before” and “after”. In non‑relativistic QM we keep a global time; in GR the parameter is replaced by proper time along world‑lines, but the idea of a parameter that orders events stays.
Momentum as generator of translations	In QM, (\hat{p}) is the operator that generates spatial translations (via (e^{-i\hat{p}x/\hbar})). In GR, the covariant momentum (p_\mu) is the generator of diffeomorphisms along a world‑line.
Energy as generator of time translations	In QM, (\hat{H}) generates time evolution (Schrödinger equation). In GR, the Hamiltonian constraint generates evolution in the chosen foliation of spacetime.
Conservation laws from symmetries	Noether’s theorem works in Lagrangian/Hamiltonian formulations that are common to all three theories.
Mass as a label of representation	In QM mass appears in the Schrödinger equation, in GR it appears in the stress‑energy tensor; both describe how a system responds to geometry or fields.

Thus the names persist because the same mathematical roles persist, even though the formulas change.

Step 3 – How each modern theory inherits the concepts

3.1 Quantum Mechanics (QM)

Newtonian concept	QM analogue	How it appears
Position (\mathbf{x})	Position operator (\hat{\mathbf{x}})	Eigenstates (	\mathbf{x}\rangle) form a continuous basis of the Hilbert space.
Momentum (\mathbf{p})	Momentum operator (\hat{\mathbf{p}} = -i\hbar\nabla)	Generates spatial translations via the unitary operator (U(\mathbf{a}) = e^{-i\mathbf{a}\cdot\hat{\mathbf{p}}/\hbar}).
Energy (E)	Hamiltonian (\hat{H})	Generates time evolution: (i\hbar\frac{d}{dt}	\psi(t)\rangle = \hat{H}	\psi(t)\rangle).
Force (\mathbf{F})	Potential term in (\hat{H}) (e.g., (\hat{V}(\hat{\mathbf{x}})))	The gradient of the potential gives the expectation value of force via Ehrenfest’s theorem: (\langle\dot{\mathbf{p}}\rangle = -\langle\nabla V\rangle).
Mass (m)	Parameter in kinetic term (\hat{T}= \hat{p}^2/2m)	Determines dispersion relation and de Broglie wavelength.
Conservation laws	Commutators with (\hat{H})	If ([\hat{O},\hat{H}]=0) then (\langle O\rangle) is constant (Noether).

Key point: The operator algebra ([\hat{x}i,\hat{p}_j]=i\hbar\delta{ij}) is a quantized version of the classical Poisson bracket ({x_i,p_j}= \delta_{ij}). Thus the classical phase‑space structure survives in a non‑commutative form.

3.2 General Relativity (GR)

Newtonian concept	GR analogue	How it appears
Space + Time (absolute)	4‑dimensional spacetime manifold (\mathcal{M}) with metric (g_{\mu\nu})	Geometry replaces the rigid background; still a stage for events.
Position (\mathbf{x}(t))	World‑line (x^\mu(\tau)) (a curve in spacetime)	Parameterised by proper time (\tau).
Momentum (\mathbf{p}=m\mathbf{v})	4‑momentum (p^\mu = m u^\mu) with (u^\mu = dx^\mu/d\tau)	Tangent to the world‑line; conserved along geodesics in symmetric spacetimes.
Energy (kinetic + potential)	Conserved quantity associated with timelike Killing vector (if spacetime has time‑translation symmetry)	(E = -p_\mu \xi^\mu) where (\xi^\mu) is the Killing vector.
Force (\mathbf{F}=m\mathbf{a})	No force; motion follows geodesic equation (u^\nu\nabla_\nu u^\mu = 0)	Acceleration is replaced by curvature of spacetime.
Mass (m)	Rest mass (invariant) appears in stress‑energy tensor (T_{\mu\nu})	Sources curvature via Einstein equation (G_{\mu\nu}=8\pi G T_{\mu\nu}).
Conservation laws	Covariant divergence (\nabla_\mu T^{\mu\nu}=0)	Generalises energy‑momentum conservation.

Key point: GR re‑expresses Newton’s ideas in a geometrical language. The same quantities (position, momentum, energy) become components of four‑vectors and tensors, but the underlying idea – a quantity that tells you “where” and “how fast” a system is – is unchanged.

Step 4 – Why the Newtonian “core” is justified

Empirical continuity – All three theories must reproduce the same experimental results in the domain where Newtonian mechanics is known to work (low speeds, weak gravity, macroscopic objects). This forces the newer theories to reduce to Newtonian formulas in the appropriate limit (the correspondence principle).
Mathematical structure – The symplectic (phase‑space) structure of classical mechanics underlies both QM (through canonical quantisation) and GR (through Hamiltonian formulations of gravity). The Lie algebra of the Galilean group (translations, rotations, boosts) is deformed into the Poincaré algebra (special relativity) and then further generalized to the diffeomorphism group (GR). The generators of these groups are still called momentum and energy.
Physical intuition – Humans experience the world in terms of where things are and how they move. The language of position, velocity, momentum, and energy is therefore an effective description that any successful theory must translate into its own formalism.
Dimensional analysis and units – The fundamental dimensions [L] (length), [M] (mass), [T] (time) remain the basis for building any physical quantity. Even in relativistic or quantum contexts, the combinations that give rise to momentum ([ML/T]) or energy ([ML^2/T^2]) are still meaningful.
Noether’s theorem – Conservation of momentum and energy follows from spatial and temporal symmetries. Those symmetries exist (though sometimes only locally) in QM and GR, so the associated conserved quantities retain the same names.

Step 5 – Summarising the “core”

The core that survives the transition can be expressed in a compact, theory‑independent way:

Core Statement
A physical system is described by a set of variables that specify its configuration (position‑like coordinates) and by conjugate variables that generate translations in those coordinates (momentum‑like quantities). If the dynamics is invariant under spatial translations, the conjugate quantities are conserved (momentum). If it is invariant under time translations, the generator of time evolution is conserved (energy). These statements hold regardless of whether the underlying geometry is Euclidean, Minkowskian, or curved, and regardless of whether the variables commute (classical) or obey quantum commutation relations.

All three theories can be written in the Hamiltonian formalism:

Newtonian: ( \dot{q}= \partial H/\partial p,\;\dot{p}= -\partial H/\partial q ) with Poisson brackets.
Quantum: ( i\hbar\dot{\hat{O}} = [\hat{O},\hat{H}] ) (Heisenberg picture).
General Relativistic (canonical) gravity: Hamiltonian constraint ( \mathcal{H}=0 ) and momentum constraints generate spacetime diffeomorphisms.

Thus the same algebraic skeleton (variables + Poisson/commutator brackets + Hamiltonian) underlies all three, and the familiar words are simply the historical labels that persist.

3. Final Answer

Answer (in plain terms):

The reason concepts such as position, momentum, and energy keep appearing in quantum mechanics and general relativity is that they are not specific to Newton’s force law; they are the structural pillars of any dynamical theory.

Position tells us the configuration of a system; in QM it becomes a operator on a Hilbert space, in GR it becomes a point on a curved spacetime manifold.
Momentum is the generator of spatial translations; the same role is played by the quantum momentum operator (-i\hbar\nabla) and by the covariant four‑momentum in GR.
Energy is the generator of time translations; it is the Hamiltonian in QM and the conserved quantity associated with a timelike Killing vector (or the Hamiltonian constraint) in GR.

These concepts survive because:

They are tied to symmetries (space‑ and time‑translation invariance) via Noether’s theorem, which holds in all three frameworks.
They appear naturally in the Hamiltonian (or Lagrangian) formulation, a mathematical language common to Newtonian mechanics, QM, and GR.
The correspondence principle forces any new theory to reproduce Newtonian results in the appropriate limit, so the same quantities must be recognizable.

Thus the “core” of Newtonian mechanics that persists is the identification of a configuration space, its conjugate momenta, and the generators of translations in space and time, together with the symmetry‑based conservation laws. This core is justified because it is rooted in universal mathematical structures (symplectic geometry, Lie algebras, Noether’s theorem) and in the empirical requirement that newer theories agree with Newtonian predictions where the latter are known to be accurate.

4. Common Mistakes

Mistake	Why it’s wrong	How to avoid it
Thinking “momentum = mass × velocity” is fundamental	In relativity the

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