List Fine Structure Conjectures

Obtain a list of numeric expressions that have been proposed as possible exact values for the fine structure constant

Use

Entity

with "PhysicalConstant" to get the value for the fine structure constant:

In[1]:=

α=

fine structure constant

PHYSICAL CONSTANT

["Value"]

Out[1]=

0.00729735257

Obtain the standard uncertainty:

In[2]:=

δα=

fine structure constant

PHYSICAL CONSTANT

["StandardUncertainty"]

Out[2]=

1.1×

-12

Express both together using

Around

In[3]:=

Around[α,δα]

Out[3]=

0.00729735256

(

)

For more than a century, scientists have come up with conjectures for closed forms. Get a list of these conjectures:

In[4]:=

conjectures=

fine structure constant

PHYSICAL CONSTANT

["ConjecturedValueAssociation"];

Format them in a table, displaying the name of the author, proposal year and proposed form:

In[5]:=

TextGrid({"Name","Year","Value"}/.#)&/@conjecturesAll,2,

//TraditionalForm

Out[5]//TraditionalForm=

Define a function to measure the complexity of a conjectured closed form in an entropy-like way through the logarithm of the size of involved integers. Mathematical functions count exponentially:

In[6]:=

complexity[0]:=0complexity[i_Integer]:=Log2[Abs[i]]complexity[r_Rational]:=complexity[Numerator[r]]+complexity[Denominator[r]]+1complexity[c_Complex]:=complexity[Re[c]]+complexity[Im[c]]+2complexity[Plus|Times]:=1complexity[_Symbol|_String]:=2complexity[expr_]:=Total[complexity/@Cases[expr,_,{-1},HeadsTrue]]

Use the function to determine the complexities and the conjectured values:

In[7]:=

dataP=Callout[{Abs[α-#Value],complexity[FunctionExpand[#Value]]},Column[{#Value,Style[Row[{"(",#Year,")"}],Gray]}]]&/@Association/@conjectures〚All,2〛;

Visualize the relationship between complexity and conjectured values. Conjectured values that are closer to the experimental value are typically more complex, and no closed form is known that is simultaneously precise and short. Examining the relationship suggests the two satisfy an inequality of the form

log(C(

))>a/log(α-

)+b+c

, where the orange curve assumes

a8

b30

, and

c1.5

In[8]:=

ShowLogLogPlotExp[8/(Log[x]+30)+1.5],{x,.6*^-12,1/100},

,ListLogLogPlotdataP,

,



Out[8]=

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Contributed by: Wolfram Staff

List Fine Structure Conjectures

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