[Translation] Error correction - physical constants in the present and new versions of the International System of Units (SI)

## Introduction

In the so-called "new SI" - the updated version of the International System of Units , which will define the seven basic units of measurement (second, meter, kilogram, ampere, kelvin, mole and candela) and which will take effect on May 20, 2019, all units SI measurements will be determined by the exact values ​​of the fundamental physical constants. As a result, all the SI units named (Newton, Volt, Ohm, Pascal, etc.) will ultimately be expressed in terms of fundamental constants. (Finally, fundamental physics will literally control our daily life ☺)

The table below shows how things will change from Monday evening, May 20 to Tuesday morning, May 21 of this year.

Calculation of data from this table is the goal of this blog. So let's start with a brief overview of what will change in the new SI.

In addition to the well-known exact value of the speed of light, in a few weeks four more physical constants — the Planck constant, the Boltzmann constant, the Avogadro constant, and the Millikan constant (more often called the elementary electric charge) —will have exact values. The decision on this change was made internationally in November last year (I wrote about it in my last blog).

Below is a fragment of page 12 of the current project SI brochures .

Note that in these definitions, decimal numbers are meant as exact decimal numbers, and not, say, machine numbers on a computer that have finite precision and are not exact numbers. The frequency of transition in the split Cs-133 hyperfine structure, the speed of light and the "light efficiency" have exact values ​​today.

## The world is discussing the coming changes

This change will have some interesting consequences for other physical constants: some constants that are currently measured and have values ​​with errors will become accurate, and some constants that currently have exact values ​​will have approximate values ​​with final errors in the future . These changes are inevitable to ensure overall system consistency.

In the first issue of this year’s Physics of the World , at letter to the editor from William Gough touched on this topic; he wrote:

With charge-on-electron ( e ) and Planck’s constant ( h ), all physical units are now “immortalized in stone”, which is very nice. But this raises an awkward question. Fine structure constant , where c is the speed of light and this is . From the familiar equations and we quickly find that . This is, of course, a pure number with a dimensionless quantity, and it is now forever fixed as equal to 1/137.13601, which is very close to the accepted value. This is not surprising, since the latter value would be used in the agreed new values ​​for e and . But nature has its own value, unknown to us at the present time, which because of this is perpetuated in a diamond. We can be forgiven for hinting that we know better than nature. But what if the future theory of the universe becomes accepted, and gives the exact value of which is significantly different from the accepted value? Could this happen? There have been attempts to find a theoretical value for α , but they include threatening and controversial quantum electrodynamics.

The problem is that in the new SI, both and will now have inaccurate values ​​with some error. In this blog, we will use the Wolfram Language and its knowledge of physical units and constants to see how these and other physical constants acquire (or lose) error, and why this is a mathematical consequence of the definition of basic units.

## Overview of the relevant ingredients of the Wolfram Language

The Wolfram Language is a unique environment for conducting numerical experiments and symbolic calculations, identifying possible consequences. In addition to the general computational capabilities, three components of the system are very useful:

1) The units of the Wolfram Language and the structure of physical quantities.

“Classic” units (such as meters, feet, etc.) can be used in calculations and visualizations. And, of course, in the translation of units of measurement.

Conversion to ordinary US units results in a fraction (and not an approximate value of a real number!) Due to precisely defined ratios of these two units.

Physicists (especially) love to use "natural" units. Often these natural units are simply physical constants or their combinations. For example, the speed of light (natural language input is used here).

Expressed in SI units (since this is speed and units are required in meters and seconds), the speed of light has an exact value.

On the other hand, the Planck constant is currently not accurate. Thus, its value, expressed in base SI units, is an approximate decimal number.

Please note that the accuracy of 6.626070 ... reflects the number of known digits.

This is the last recommended value for Planck's constant, published in CODATA 2017 to prepare for the specification of constants. Below is the corresponding table:

Physical constants (or combinations thereof) that connect two physical quantities can often be used as natural units. The simplest examples would be measuring speed in terms of the speed of light or microscopic torque in terms of . Or energy can be measured in terms of mass with an implied multiplier . The DimensionalCombinations feature can be used to search for combinations of physical constants that allow you to link two specified physical quantities. For example, the following relationship between mass and energy can be built:

The first equality reflects Einstein's famous formula , the second is the equivalent of the first relation, and the third - (dimensionally) states that .

2) The entity class " PhysicalConstant ", recently added to the Knowledge Base Wolfram Knowledgebase .

Functions and objects in the Wolfram Language language are “generated computationally,” that is, they are ready for use in calculations as well. But to describe and simulate the real world, we need data about the real world. Entity structure is a convenient and fully integrated way to obtain such data. Below is some information about the electron, proton and neutron.

One of the new entity units is physical constants. Currently, the Knowledge Base contains more than 250 physical constants.

Below are a dozen randomly selected examples. And without a clear definition of what exactly the physical constants are, the masses of the fundamental particles, the Lagrange parameters of the standard model, etc. For convenience, the list also contains astronomical constant according to the Astronomical Almanac.

Most of the fundamental physical constants were called class C constants in the famous work by Jean-Marc Levy-Leblond. The following are the constants of classes C and B.

Take, for example, the natural unit of time, the time of Planck. Functions ToEntity and FromEntity make it easy to go back and forth between physical constants as units and physical constants as entities. The following is an entity that corresponds to a Planck time unit.

The knowledge base has a lot of meta-information about it, for example, its values ​​in the latest CODATA lists.

The last conclusion, which contains the value and error, brings us to a third important function that will be useful later:

3) Introduction of the Around function [] in version 12 of the Wolfram Language. The Around [] function provides an inaccurate value, indicating the mean and error. The Around [] arithmetic model is based on a GUM (Error Expression Guide in measurements) - do not confuse with Leibniz Plus-Minus-calculus. Here is a value with an error.

The most important and useful aspect of calculations with values ​​that have inaccuracies is that they properly take into account correlations. The naive use of such values ​​in arithmetic numbers or intervals may underestimate or overestimate the error that occurs.

We can see below that the AroundReplace [] function takes into account the correlation.

## Let's go back to the letter to the editor

Now let's use these three components and the William Gough letter to the editor in more detail.

At current approximations for e and , these two values ​​for the fine structure constant are consistent within their errors.The first is the expression from the letter to the editor, and the second is the quantity ( Quantity []), representing the constant thin structures.

Every few years, CODATA publishes the official values ​​of the fundamental constants (see fine structure constants ); as I said, the values ​​used in the Wolfram Language are the last CODATA values ​​and the final error is reflected in the exact numbers.

Please note that the directly measured value of the fine structure constant is slightly more accurate than that which expresses the fine structure constant through other constants.

If we use the upcoming exact values ​​ e and , and we use the current exact value we get the following exact value for the fine structure constant in the form .

It is unlikely that a Lord who does not even play dice would choose such a number for in our universe. This means that while e and will be fixed in the new SI, the current exact values ​​> and must inevitably be“ fixed ”(see also article Goldfarb about the value of new SI). (We’ll go back to why and should soon become inaccurate.)

This means that after May 20 of this year, these results will be different from those shown below.

(In a brief note, the entity class " PhysicalConstant " also has assumed values ​​for constants, such as the fine structure constant):

Now, apart from the theological argument about the exact form of the fine structure constant, from a physical point of view, why and should be inaccurate? As an argument of probability let's consider . One of the most outstanding results is Coulomb's law.

In the existing SI system, the amp has a "precise" definition:

Amp is a constant current which, if maintained in two parallel conductors of infinite length with a slight circular cross section and placed at a distance of 1 meter in a vacuum, will create between these conductors a force equal to newton per meter length.

This definition uses the purely mechanical quantities Newton and Meter (that is, after expansion, it is a second, meter, and kilogram). No relationship with the electron charge is made, and in the existing SI system, the elementary charge is an experimentally measured quantity.

And this experimentally measured value has changed over the years becoming more accurate.

The force in the left part of the Coulomb's law (expressed in Newtons) contains the basic unit kilogram, which, after the value of Planck's constant becomes constant, also becomes precisely definable. Since there is no reason to assume that all laws of nature can be expressed in finite rational numbers, the only possible "moving part" in Coulomb's law would be . Its numerical value must be determined, and it will make the left and right side of Coulomb's law coincide.

From a more fundamental point of view of physics, the fine structure constant is an interaction constant that determines the strength of electromagnetic interactions. And, perhaps, one day physics will be able to calculate the value of the fine structure constant, but we are still far from that. Just choosing the definitions of units cannot fix the value .> cut>

Are both and become non-fixed, or is it possible to keep one of them accurate? Because of the already accurate speed of light and the ratio , if one of img alt = "image" src = "https://habrastorage.org/getpro/habr/post_images/033/d20/e90/033d20e90431b22c0ac3e4f0cd8a5e86.png"/> or exact, the other must also be accurate. We know that at least one must become uncommitted, so it follows that both must be uncommitted.

The values ​​that are now given by Planck’s constant, Boltzmann’s constant, Avogadro’s constant and the elementary charge are neither arbitrary nor fully defined.They are defined up to about eight characters, so that the units that they define after May 20 correspond to the “size” of the units that they define before May 20. But the numbers on the bottom right are not defined. Thus, the value of the future exact value of the elementary charge can be , and not . This is Occam's razor and rationality that let us use .

At a more technical level, the substitution in the previous calculation was that through the term in the formula the amp was used before the override (remember ), but the exact value of the elementary charge was also used meaning the definition of an ampere after the override. And we always need to stay in the same system of units.

## Computing a table of error-optimized forms

So, the natural question arises: what should these “non-fixed” values ​​be? In my last blog, I manually created a new value . What can be done manually can be done using a computer program, so let's implement a small program that calculates the form of derivatives of physical constants optimized by error. In a future-oriented approach, an entity class of seven constants that defines a new SI is already available.

Below are the constants that will have the exact value in the new SI.

The current values ​​of these constants together with their error (calculated using the Around [] function) are:

Using the entity class " PhysicalConstant " we can get new, upcoming values ​​of physical quantities. Note that, as in all computer languages, exact integers and rational numbers are either explicit integers or rational (but not decimal) numbers.

Many physical constants can be related by equations defined by physical theories of various fields of physics. In the future, we want to limit ourselves to the theory of fundamental electromagnetic phenomena, in which the error of the constants will be reduced to the error of the fine structure constant and Rydberg's constant If we included, for example, gravitational phenomena, we would have to use the gravitational constant G, which is measured independently, although it has a very large error (which is why the NSF had the so-called Big-G Challenge ").

Next, we confine ourselves to electrical, magnetic, and mass quantities, the errors of which are reduced to units and .

Below we use the new Around function to express values ​​with corresponding errors and .

Currently, according to CODATA 2014, the relative error for is about , and for about . As you can see, the error for is greater than for .

Below is a graph of the log-base-10 relative error as a function of a and b . Obviously, for small degrees, the relative error of the product weakly depends on the exponents a and b . This graph shows that the dependence of the error is dominant relative to i > a (a measure of the degree of fine structure). This observation is explained by the fact that the error of the Rydberg constant is 50 times less than the error of the fine structure constant.

To calculate the errors of different constants in the new SI, we will use the following steps:

• Extract equivalent representations for physical constants that are accessible from the “ PhysicalConstant ” entity class.

These identical equalities between physical constants are laws of physics and, as such, should be preserved both in the old and in the new SI.

• Perceive formulas as a set of algebraic equalities to which various exclusion methods can be applied to express a constant through a combination of the seven basic constants of the new SI, as well as the fine structure constants and Rydberg's constants .

These are the nine basic constants that we allow to apply in the definitions of each new constant under consideration. (Technically, there are 10 constants in the list, but due to the simple scaling relation between h and , there are actually nine "different" constants in this list .)

The entity class “ PhysicalConstant ” contains a lot of information about the relationship between physical constants. For example, here are the equivalent forms of the four constants that are currently being measured and will soon be identified as having exact values.

Within the limits of the accuracy of the measured values, all these individual elements work now. Here is a quick numerical check of alternative forms of Planck’s constant. But the specific numerical value, especially the error, depends on the actual form of the representation. Using Around [], we can easily calculate the resulting errors.

Below is a graphical representation of the resulting errors of various representations. A very large error of dozens of representations can be traced to a large error in the second constant radiation.

And again, in the framework of the error of constants, this relationship should be maintained after the redefinition. So which of these representations is best used after the override to minimize errors? Perhaps none of these equivalents is optimal, and by combining some of these representations, you may be able to build a better one that has a smaller resultant error.

Now for the next steps of algebraic exceptions, we convert the constants found in the equivalent classes (this is possible because the second arguments are in Entity [" PhysicalConstant",. ] and in Quantity [< b> 1,. ] equated). The reason why we use entities, rather than values ​​in subsequent calculations, is twofold: first, entities are convenient, easily readable representations; and secondly, algebraic functions (such as GroebnerBasis ) do not penetrate into the values ​​to determine the nature of their first argument.

Then we make all identities polynomials.The last step means: (1) subtracting the left side from the right side; and (2) that no fractional powers (for example, square roots) of the constants no longer appear. Such a transformation into a polynom we carry out by finding all fractional exponents and finding the LCM (lowest common factor) of all their denominators.

Below is one of the previous equations that contains constants with fractional degrees.

After polynomialization, we arrive at polynomials of several variables in the three constants present. These polynomials must be eliminated.

The following table shows how the toPolynomial function applies to the equivalent forms shown earlier for elementary charge. After canonization, ℏ in some of the resulting polynomials become identical.

Now, based on the existing physical constants (without the constants used in the definition of the new SI), we get enough equivalent forms to create a set of equations.

Below is a list of the polynomial equations obtained to express an elementary charge.

Express all errors through errors and = "https://habrastorage.org/getpro/habr/post_images/ee5/3fa/1d8/ee53fa1d8d9c909ac15708476a98ba20.png"/>. Only these two constants are enough to express the error of many physical constants. And since their errors do not depend on each other, and since errors are rather small, these two fairly well-known constants are best suited to express the optimized (in terms of errors) new version of many physical constants. And, of course, we allow all seven exact constants from the new SI; since they are exact values, their presence will not change the error.

The main work on expressing a given constant in SI constants and and will be implemented by the GroebnerBasis function. Setting the MonomialOrder - & gt; ElventionOrder is a critical step that removes all “unnecessary” physical quantities, leaving one polynomial equation with precisely defined constants and (if necessary) fine structure and Rydberg constants.

By eliminating the electron-related constants, we get .

The error of this expression arises from the member . We can define a function that removes the member causing an error.

For a more compact representation, we can define a function that returns an equivalent form, as well as old and new errors - in the form of a string.

We end the blog with a table of old and new errors for more than a dozen physical constants. This list is chosen as a representative example; other constants can be processed in the same way (this may require adding additional inaccurate constants to be preserved, such as the gravitational constant or standard model parameters).

Combining rows into a table gives the following result for the optimal representation of these constants in the new SI unit of units.

It was a table that we intended to display, and we managed to derive it. Notice the appearance of in the numerator and denominator and in such a way that the result is the ac. A similar list can be found at the bottom of the Wikipedia page on redefining SI units .

Now we can calmly expect World Metrology Day 2019 for a fundamentally better world described through fundamental constants.