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ASlateff
ASlateff
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There are already good answers for Q1-Q3, so I'll clarify Q4.

In physics, quantities typically have a dimension, such as length, mass, time etc., and a unit, such as metre, kilogram, second, etc. Pure mathematical numbers have neither a dimension nor a unit.

Let's consider time. A possible mathematical model for time is to use a one-dimensional oriented affine space $\mathbb{T}$ over the real numbers. By choosing an origin ("time zero") $O \in \mathbb{T}$ and a ("positive") reference time $t_1 \in \mathbb{T}\setminus O$, such that $t_1$ appears after $O$ with respect to the orientation, the pair $(O, t_1)$ provides an oriented affine coordinate base. This coordinate base induces an isomorphism of oriented real affine spaces $\mathbb{T} \to \mathbb{A}_\mathbb{R}^1$, where the real affine space $\mathbb{A}_\mathbb{R}^1$ carries the orientation induced by the real numbers. In particular, we map $O \mapsto 0$ and $t_1 \mapsto 1$.

This isomorphism of real affine spaces also induces an isomorphism of the corresponding real vector spaces of oriented time intervals (durations). In $\mathbb{T}$, the vector $T_{ref}$ with $O + T_{ref} = t_1$ gives a reference unit $T_{ref}$ for durations. Given a duration $T$, we obtain a real number $t$ by $t := T/T_{ref}$. In this sense, the reference unit $T_{ref}$ provides a time scale.

An equation of time intervals can be made dimensionless by dividing each occurring time by the reference time $T_{ref}$ and also each variable of a duration by $T_{ref}$.

Example. If thewe are given an equation for time events, which is $$2 s + T = 5 s,$$ and if my reference time length is $T_{ref} = 1 s$, then I will declare $t := X/T_{ref}$ and I will obtain in dimension-less form $$2 + t = 5.$$ This equation basically has the same structure and the same look, but it is dimensionless. It is an equation for real numbers only.

Now, it is often reasonable to have physical processes, where different choices of a good reference time are appropriate. Say, we are given an atomic process, where a good reference time length is $T_1 := 10^{-12} s$, and a geological process, where a good reference time length is $T_2 := 10^{12} s$. If we make times dimensionless by $t := T/T_1$, then the other option would be to consider $T/T_2 = T_1 / T_2 \cdot T/T_1 = \varepsilon \cdot t$ with a parameter $$\varepsilon := T_1/T_2 \ll 1$$ which is (positive and) very small. So, if we make atomic times dimensionless by considering $t$, and if me make geological times dimensionless by considering $\varepsilon t$, then the small parameter $\varepsilon$ will occur in our mathematical equations.

A possible treatment would perhaps be to apply a perturbation method, where we take formally $\varepsilon \to 0+$ in the equations to obtain a reduced problem, which might be better treatable, and then try an expansion ansatz in $\varepsilon$. The limit $\varepsilon \to 0+$ is only a mathematical limit. As for the physical problem, it means, that the quotient of the choosen reference times $T_1/T_2 \to 0+$, because $T_1 \ll T_2$, in order to obtain a proper reduced physical problem.

Basically, the same thing happens with "$\hbar \to 0+$." One chooses a reference length, reference time, and reference mass, and bringsone rewrites the equations in a dimensionless form. If If we have another good reference time (or reference length or reference mass), then a small parameter $\varepsilon$ will appear, maybe in exactly the same positions, where $\hbar$ has occurred in the original physical equations. So one simply writes $\hbar$ instead of $\epsilon$, to keep the notational form of the original physical equations, but in fact, dimensionless pure mathematical equation are meant, that now contain an additional small parameter.

There are already good answers for Q1-Q3, so I'll clarify Q4.

In physics, quantities typically have a dimension, such as length, mass, time etc., and a unit, such as metre, kilogram, second, etc. Pure mathematical numbers have neither a dimension nor a unit.

Let's consider time. A possible mathematical model for time is to use a one-dimensional oriented affine space $\mathbb{T}$ over the real numbers. By choosing an origin ("time zero") $O \in \mathbb{T}$ and a ("positive") reference time $t_1 \in \mathbb{T}\setminus O$, such that $t_1$ appears after $O$ with respect to the orientation, the pair $(O, t_1)$ provides an oriented affine coordinate base. This coordinate base induces an isomorphism of oriented real affine spaces $\mathbb{T} \to \mathbb{A}_\mathbb{R}^1$, where the real affine space $\mathbb{A}_\mathbb{R}^1$ carries the orientation induced by the real numbers. In particular, we map $O \mapsto 0$ and $t_1 \mapsto 1$.

This isomorphism of real affine spaces also induces an isomorphism of the corresponding real vector spaces of oriented time intervals (durations). In $\mathbb{T}$, the vector $T_{ref}$ with $O + T_{ref} = t_1$ gives a reference unit $T_{ref}$ for durations. Given a duration $T$, we obtain a real number $t$ by $t := T/T_{ref}$. In this sense, the reference unit $T_{ref}$ provides a time scale.

An equation of time intervals can be made dimensionless by dividing each occurring time by the reference time $T_{ref}$ and also each variable of a duration by $T_{ref}$.

Example. If the equation is $$2 s + T = 5 s,$$ and if my reference time length is $T_{ref} = 1 s$, then I will declare $t := X/T_{ref}$ and I will obtain in dimension-less form $$2 + t = 5.$$ This equation basically has the same structure and the same look, but it is dimensionless. It is an equation for real numbers only.

Now, it is often reasonable to have physical processes, where different choices of a good reference time are appropriate. Say, we are given an atomic process, where a good reference time length is $T_1 := 10^{-12} s$, and a geological process, where a good reference time length is $T_2 := 10^{12} s$. If we make times dimensionless by $t := T/T_1$, then the other option would be to consider $T/T_2 = T_1 / T_2 \cdot T/T_1 = \varepsilon \cdot t$ with a parameter $$\varepsilon := T_1/T_2 \ll 1$$ which is (positive and) very small. So, if we make atomic times dimensionless by considering $t$, and if me make geological times dimensionless by considering $\varepsilon t$, then the small parameter $\varepsilon$ will occur in our mathematical equations.

A possible treatment would perhaps be to apply a perturbation method, where we take formally $\varepsilon \to 0+$ in the equations to obtain a reduced problem, which might be better treatable, and then try an expansion ansatz in $\varepsilon$. The limit $\varepsilon \to 0+$ is only a mathematical limit. As for the physical problem, it means, that the quotient of the choosen reference times $T_1/T_2 \to 0+$, because $T_1 \ll T_2$, in order to obtain a proper reduced physical problem.

Basically, the same thing happens with "$\hbar \to 0+$." One chooses a reference length, reference time, and reference mass and brings the equations in a dimensionless form. If we have another reference time (or reference length), then a small parameter $\varepsilon$, maybe in exactly the same positions, where $\hbar$ has occurred in the original physical equations. So one simply writes $\hbar$ instead of $\epsilon$, to keep the notational form of the original physical equations, but in fact, dimensionless pure mathematical equation are meant, that now contain an additional small parameter.

There are already good answers for Q1-Q3, so I'll clarify Q4.

In physics, quantities typically have a dimension, such as length, mass, time etc., and a unit, such as metre, kilogram, second, etc. Pure mathematical numbers have neither a dimension nor a unit.

Let's consider time. A possible mathematical model for time is to use a one-dimensional oriented affine space $\mathbb{T}$ over the real numbers. By choosing an origin ("time zero") $O \in \mathbb{T}$ and a ("positive") reference time $t_1 \in \mathbb{T}\setminus O$, such that $t_1$ appears after $O$ with respect to the orientation, the pair $(O, t_1)$ provides an oriented affine coordinate base. This coordinate base induces an isomorphism of oriented real affine spaces $\mathbb{T} \to \mathbb{A}_\mathbb{R}^1$, where the real affine space $\mathbb{A}_\mathbb{R}^1$ carries the orientation induced by the real numbers. In particular, we map $O \mapsto 0$ and $t_1 \mapsto 1$.

This isomorphism of real affine spaces also induces an isomorphism of the corresponding real vector spaces of oriented time intervals (durations). In $\mathbb{T}$, the vector $T_{ref}$ with $O + T_{ref} = t_1$ gives a reference unit $T_{ref}$ for durations. Given a duration $T$, we obtain a real number $t$ by $t := T/T_{ref}$. In this sense, the reference unit $T_{ref}$ provides a time scale.

An equation of time intervals can be made dimensionless by dividing each occurring time by the reference time $T_{ref}$ and also each variable of a duration by $T_{ref}$.

Example. If we are given an equation for time events, which is $$2 s + T = 5 s,$$ and if my reference time length is $T_{ref} = 1 s$, then I will declare $t := X/T_{ref}$ and I will obtain in dimension-less form $$2 + t = 5.$$ This equation basically has the same structure and the same look, but it is dimensionless. It is an equation for real numbers only.

Now, it is often reasonable to have physical processes, where different choices of a good reference time are appropriate. Say, we are given an atomic process, where a good reference time length is $T_1 := 10^{-12} s$, and a geological process, where a good reference time length is $T_2 := 10^{12} s$. If we make times dimensionless by $t := T/T_1$, then the other option would be to consider $T/T_2 = T_1 / T_2 \cdot T/T_1 = \varepsilon \cdot t$ with a parameter $$\varepsilon := T_1/T_2 \ll 1$$ which is (positive and) very small. So, if we make atomic times dimensionless by considering $t$, and if me make geological times dimensionless by considering $\varepsilon t$, then the small parameter $\varepsilon$ will occur in our mathematical equations.

A possible treatment would perhaps be to apply a perturbation method, where we take formally $\varepsilon \to 0+$ in the equations to obtain a reduced problem, which might be better treatable, and then try an expansion ansatz in $\varepsilon$. The limit $\varepsilon \to 0+$ is only a mathematical limit. As for the physical problem, it means, that the quotient of the choosen reference times $T_1/T_2 \to 0+$, because $T_1 \ll T_2$, in order to obtain a proper reduced physical problem.

Basically, the same thing happens with "$\hbar \to 0+$." One chooses a reference length, reference time, and reference mass, and one rewrites the equations in a dimensionless form. If we have another good reference time (or reference length or reference mass), then a small parameter $\varepsilon$ will appear, maybe in exactly the same positions, where $\hbar$ has occurred in the original physical equations. So one simply writes $\hbar$ instead of $\epsilon$, to keep the notational form of the original physical equations, but in fact, dimensionless pure mathematical equation are meant, that now contain an additional small parameter.

Source Link
ASlateff
ASlateff
  • 96
  • 4

There are already good answers for Q1-Q3, so I'll clarify Q4.

In physics, quantities typically have a dimension, such as length, mass, time etc., and a unit, such as metre, kilogram, second, etc. Pure mathematical numbers have neither a dimension nor a unit.

Let's consider time. A possible mathematical model for time is to use a one-dimensional oriented affine space $\mathbb{T}$ over the real numbers. By choosing an origin ("time zero") $O \in \mathbb{T}$ and a ("positive") reference time $t_1 \in \mathbb{T}\setminus O$, such that $t_1$ appears after $O$ with respect to the orientation, the pair $(O, t_1)$ provides an oriented affine coordinate base. This coordinate base induces an isomorphism of oriented real affine spaces $\mathbb{T} \to \mathbb{A}_\mathbb{R}^1$, where the real affine space $\mathbb{A}_\mathbb{R}^1$ carries the orientation induced by the real numbers. In particular, we map $O \mapsto 0$ and $t_1 \mapsto 1$.

This isomorphism of real affine spaces also induces an isomorphism of the corresponding real vector spaces of oriented time intervals (durations). In $\mathbb{T}$, the vector $T_{ref}$ with $O + T_{ref} = t_1$ gives a reference unit $T_{ref}$ for durations. Given a duration $T$, we obtain a real number $t$ by $t := T/T_{ref}$. In this sense, the reference unit $T_{ref}$ provides a time scale.

An equation of time intervals can be made dimensionless by dividing each occurring time by the reference time $T_{ref}$ and also each variable of a duration by $T_{ref}$.

Example. If the equation is $$2 s + T = 5 s,$$ and if my reference time length is $T_{ref} = 1 s$, then I will declare $t := X/T_{ref}$ and I will obtain in dimension-less form $$2 + t = 5.$$ This equation basically has the same structure and the same look, but it is dimensionless. It is an equation for real numbers only.

Now, it is often reasonable to have physical processes, where different choices of a good reference time are appropriate. Say, we are given an atomic process, where a good reference time length is $T_1 := 10^{-12} s$, and a geological process, where a good reference time length is $T_2 := 10^{12} s$. If we make times dimensionless by $t := T/T_1$, then the other option would be to consider $T/T_2 = T_1 / T_2 \cdot T/T_1 = \varepsilon \cdot t$ with a parameter $$\varepsilon := T_1/T_2 \ll 1$$ which is (positive and) very small. So, if we make atomic times dimensionless by considering $t$, and if me make geological times dimensionless by considering $\varepsilon t$, then the small parameter $\varepsilon$ will occur in our mathematical equations.

A possible treatment would perhaps be to apply a perturbation method, where we take formally $\varepsilon \to 0+$ in the equations to obtain a reduced problem, which might be better treatable, and then try an expansion ansatz in $\varepsilon$. The limit $\varepsilon \to 0+$ is only a mathematical limit. As for the physical problem, it means, that the quotient of the choosen reference times $T_1/T_2 \to 0+$, because $T_1 \ll T_2$, in order to obtain a proper reduced physical problem.

Basically, the same thing happens with "$\hbar \to 0+$." One chooses a reference length, reference time, and reference mass and brings the equations in a dimensionless form. If we have another reference time (or reference length), then a small parameter $\varepsilon$, maybe in exactly the same positions, where $\hbar$ has occurred in the original physical equations. So one simply writes $\hbar$ instead of $\epsilon$, to keep the notational form of the original physical equations, but in fact, dimensionless pure mathematical equation are meant, that now contain an additional small parameter.