Abstract
In this chapter we start with a discussion of a fundamental assumption on the nature of matter, namely the idea of particles as basic constituents and the consequences thereof. In Sects. 2.1–2.3 we focus on clarifying some terminology in computational multiscale modeling and learn about the concepts of “models” in science. Section 2.4 focuses on hierarchical modeling concepts in classical and quantum physics which is followed by discussing the standard model of elementary particle physics and the attempts of reducing all understanding of material behavior to the interactions of some basic constituents (particles). In Sect. 2.6 we introduce the basics of computer science such as recursive programming, the concepts of formal (computer) languages, finite automata, complexity theory and the Turing machine. The chapter ends with some reading suggestions and several problems as an offer for the ambitious reader.
I think there is a world market for maybe five computers.
Thomas Watson, IBM chairman, 1943
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- 1.
From Greek “\( \acute{\upalpha }\tau \text {o}\upmu \text {o}\)” (indivisible).
- 2.
The original publication is [72]. For a review, originally published in 1856 and available today as unabridged facsimile, see [73].
- 3.
In contrast to Ernst Mach – roughly half a century later – Richard Feynman starts the first chapter of his famous lecture series on physics [75] with the remark that the atomic hypothesis, i.e. the idea that matter is made of single small particles, contains the most information on the world with the least number of words.
- 4.
This peculiar naming of the currently smallest known constituents of matter after the sentence “Three quarks for Master Mark” that appears in James Joyce’s novel “Finnegan’s Wake”, goes back to Murray Gell-Mann (Nobel price 1969).
- 5.
This property of the strong interaction was discovered by D. Gross, D. Politzer, and F. Wilczek (Nobel prize 2004) and is due to an increase of the strong coupling constant (and along with it an increase of the strong force) with increasing distance of the quarks. That is, if one tries to separate quarks, energy is “pumped” into the force field until – according to \(E=mc^2\) – quark-anti-quark systems come into being. The original publications are [77–79].
- 6.
Also compare Fig. 7.23 on p. 349.
- 7.
The refractive index \(n=1\) in the vacuum of an electron microscope.
- 8.
Solipsism in this context is not a scientific category, as it renders all rational discussions useless and impossible.
- 9.
Of course, this is only true, if the object has a velocity component parallel to the surface of Earth.
- 10.
Time-reversibility (or time-symmetry) is actually broken in certain rare elementary particle physics processes which was shown in 1964 by J.L. Cronin and V.L. Fitch (Nobel Prize 1980) at CERN.
- 11.
For example, it has never been observed, that the pieces of a broken cup cool off and repair themselves, although this process is not forbidden by a fundamental law of nature.
- 12.
After the 14th century monk William of Occam who often used the principle of unnecessary plurality of medeval philosophy in his writings, such that his name eventually became connected to it.
- 13.
See e.g. the comments on p. 54 of Steven Weinberg’s book “Dreams of a final theory” [91].
- 14.
Albert Einstein’s quest for a formulation of general relativity during the years 1907–1915 is the classic example, cf. Chap. 3 on p. 109.
- 15.
\({R_{ijkl}}_{\arrowvert m} + {R_{ijlm}}_{\arrowvert k} + {R_{ijmk}}_{\arrowvert l}=0\), see e.g. [96].
- 16.
In fact, the N-body problem is analytically unsolvable for \(N\ge 3\), cf. Example 37 on p. 258.
- 17.
In continuum theory this is the antisymmetric part of the tensor of the displacement derivatives.
- 18.
After the mathematician Hermann Minkowski , mathematics professor at the Polytechnikum ETH in Zurich and teacher of Einstein, who introduced this unifying concept in a famous lecture at Cologne in 1908, see pp. 54–71 in [116].
- 19.
A vector is just a first order tensor.
- 20.
The most prominent application of this kinematic effect is the twin paradox, see e.g. [121, 122], which is based on the asymmetry between the twin that stays in the same IS at home, and the one who travels, and has to accelerate, i.e. to switch inertial systems, in order to return. For a modern treatment of the clock paradox, see [122].
- 21.
Non-invariancy of equations upon coordinate transformations.
- 22.
See Sect. 3.3.8 on p. 153.
- 23.
Thus in a sense, the term “theory of relativity” is a misnomer and it had probably better be called “theory of invariants”.
- 24.
For a definition of “algorithm”, see Sect. 2.6.
- 25.
For the definition of an Abelian group, see Box 2.1 on p. 68.
- 26.
The positron \(e^+\) is the anti-particle of the electron \(e^-\).
- 27.
For example, a coherent laser beam comprises billions of photons oscillating in a single state.
- 28.
The three shared the Nobel prize in 1965.
- 29.
In Dirac’s hole theory, a “vacuum” is interpreted as the negative energy spectrum of the solutions of his equation. The holes in the “Dirac sea” of negative energies were first interpreted by Dirac as protons [134], but this idea was quickly abandoned under the impression of several arguments put forward by W. Pauli and others.
- 30.
For a discussion of the importance of manifolds in modern physical theory, see Chap. 3.
- 31.
If one were to rename gauge fields today, they would probably be called “phase fields” as the symmetry with matter fields is with respect to their phase, not to some length scale.
- 32.
Hyperons are baryons with a strangeness quantum number.
- 33.
For example, \(T^{ab}{}_{[cd]}= \frac{1}{2}(T^{ab}{}_{cd} - T^{ab}{}_{dc})\).
- 34.
www.expasy.ch/sprot.
- 35.
Diophantic equations are polynomial equations with integer coefficients for which an integer solution is sought. This problem is also known as Hilbert’s 10th problem which was raised by him in the year 1900 [172]. It was not before 1970, when Hilbert’s 10th problem could be proved to be unsolvable by Yuri Matiyasevič.
- 36.
A simple example for an algorithm that has as input a different algorithm is a compiler.
- 37.
- 38.
This is one of the great unsolved problems in mathematics. The Clay Mathematics Institute in Cambridge, MA, U.S.A. is offering a 1 million $ reward to anyone who has a formal proof that \(\mathbf{P} = \mathbf{NP}\) or that \(\mathbf{P}\ne \mathbf{NP}\).
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Problems
Problems
Problem 1
Proper Time Interval \(\mathrm{d}\tau \)
Show that the Lorentz-transformations \(\varLambda ^{\alpha }_{\beta }\) leave the proper time \(\mathrm{d}\tau \) (see p. 52) invariant.
Problem 2
Conservation Laws
State for each of the following particle reactions whether it is forbidden or not. If applicable, state the conservation law that is violated.
-
(a)
\(\bar{p} + p \rightarrow \mu ^+ + e^-\,,\)
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(b)
\(n\rightarrow p + e^- + \nu _e\,,\)
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(c)
\(p\rightarrow n + e^+ + \nu _e\,.\)
Push (a) and pop (b) operation with stacks
Problem 3
Euler-Lagrange Equations
Perform the variation of (2.40) and show that (2.41) are the corresponding equations of motion.
Problem 4
Klein–Gordon and Dirac Equation
Show that the field \(\psi \) of (2.62) on p. 73 satisfies the correct energy-momentum relation, i.e. it satisfies the Klein–Gordon equation (2.57). Derive from this a set of equations for the \(\alpha _i\) and \(\beta \).
Problem 5
Abstract Data Types: LIFO Structure
The two basic operations of stacks (LIFO structures) are push (putting one data element on the stack) and pop, cf. Fig. 2.34.
Write an implementation of the stack with a push and pop functionality in C++ using a modular design, i.e. use a header file “Stack.h” for declarations, a file “Stack.cpp” and a main procedure which tests this implementation.
Problem 6
An implementation of the Ackermann function
Write a recursive implementation of the Ackermann function (2.92a), (2.92b), (2.92c). How long does it take to compute A(5, 0)? (You can go and drink a coffee in the mean time). What about A(5, 1)?
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Steinhauser, M.O. (2017). Multiscale Computational Materials Science. In: Computational Multiscale Modeling of Fluids and Solids. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53224-9_2
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