The Principles of Quantum Theory, From Planck's Quanta to the Higgs Boson: The Nature of Quantum Reality and the Spirit of Copenhagen

341 Pages·2016·2.46 MB·English

by  Arkady Plotnitsky

Arkady Plotnitsky
The Principles of
Quantum Theory,
From Planck's
Quanta to the Higgs
Boson
The Nature of Quantum Reality and the
Spirit of Copenhagen
The Principles of Quantum Theory,
From Planck’s Quanta to the Higgs Boson
Arkady Plotnitsky
The Principles of Quantum
Theory, From Planck’s
Quanta to the Higgs Boson
The Nature of Quantum Reality
and the Spirit of Copenhagen
Arkady Plotnitsky
Theory and Cultural Studies Program
Purdue University
West Lafayette, IN, USA
ISBN 978-3-319-32066-3 ISBN 978-3-319-32068-7 (eBook)
DOI 10.1007/978-3-319-32068-7
Library of Congress Control Number: 2016940380
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I am working out a quantum theory about it
for it is really most tantalizing state of
affairs.
—James Joyce, Finnegans Wake
(Joyce 2012, p. 149)
Preface
I would like to begin with the endpoint of the history to be traversed by this study,
the discovery of the Higgs boson, arguably the greatest event of fundamental phys-
ics in the twenty-first century thus far, and, thus far, a culminating event in the his-
tory of quantum physics. This discovery has been discussed at all levels and in all
media, with photographs of the “events” testifying to the existence of the Higgs
boson and of various components, staggering in their complexity, of the Large
Hadron Collider (LHC), and the relevant parts of the mathematical formalism of
quantum field theory (e.g., “The Higgs Boson,” Wikipedia; CERN: Accelerated
Science: Images). These pictures are well known and easily located on the Web. I
only cite the key part of the formalism, the epistemological nature of which will be
discussed in Chap. 6:
In the Standard Model, the Higgs field is a four component scalar field that forms a complex
doublet of the weak isospin SU(2) symmetry:
1 æf1+if2ö
f= ç ÷
2 èf0 +if3ø

while the field had charge +1/2 under the weak hypercharge U(1) symmetry (in the conven-
tion where the electric charge, Q, the weak isospin, I, and the weak hypercharge, Y, are
3
related by Q = I + Y).
3
The Higgs part of the Lagrangian is
2
 =æç¶ -igWata-ig¢B ö÷ +m2f†f–l(f†f)2,
H è m m 2 mø

where Wμα and Bμ are the gauge bosons of the SU(2) and U(1) symmetries, and g and g′ their
respective coupling constant, ta =sa/2 (where σα are the Pauli matrices) a complete set
of generators of the SU(2) symmetry, and l>0 andm2>0, so that the ground state
breaks the SU(2) symmetry. The ground state of the Higgs field (the bottom of the potential)
vii
viii Preface
is degenerate with different ground states related to each other by an SU(2) gauge transfor-
mation. It is always possible to pick up a gauge such that the ground state f1 =f2 =f3 =0.
The expectation value of ϕ0 in the ground state (the vacuum expectation value or vev) is
v m 246GeV
then f0 = , where v= . The measured value of this parameter is ~ .
Ö2 Öl c2
It has units of mass, and is the only free parameter of the Standard Model that is not a
dimensionless number. Quadratic terms W and B arise, which give masses to the W and Z
μ μ
bosons:
v g
M =
W 2

g2 +g¢2
M =
Z 2

M g
with their ration determining the Weinberg angle, cosq = W = , and
leave a massless U(1) photon, γ. W MZ g2 +g¢2
(“The Higgs Boson,” Wikipedia; Peskin and Schroeder 1995, pp. 690–700)
Now, what does all this (the photographs of the corresponding events, computer
generated images and data, staggering machinery of the LHC, and the mathematics
just described) mean? And how is it possible? Without attempting to definitively
answer these questions, this study will consider a particular perspective on them,
indeed a particular way of asking them, and will suggest partial answers that arise if
one adopts this perspective. This perspective is guided by understanding the nature
of quantum reality, or the quantum reality of nature, and of quantum theory, from
quantum mechanics to quantum field theory, in “the spirit of Copenhagen
[Kopenhagener Geist der Quantenheorie],” in Heisenberg’s memorable phrase, the
spirit that guides this study, as indicated by its subtitle (Heisenberg 1930, p. iv). This
understanding relates nature and spirit (a relation that we seem unable to do without
even when a materialist view of the world is adopted) in a new way. The spirit of
Copenhagen, I argue, is defined by three great divorces from the preceding under-
standing of these relationships between nature and spirit, or, to use a less theologi-
cally charged expression, nature and mind (technically, German Geist means both),
specifically scientific thought in modern physics: reality from realism, probability
from causality, and locality from relativity. It is true that the last of these divorces
did not shape the rise of the spirit of Copenhagen in the way the first two did, but it
became a major part of this spirit nevertheless.
I shall comment on these three “divorces” and define the corresponding concepts
below, and discuss them in detail in Chap. 1 and elsewhere in this study. For the
moment, the spirit of Copenhagen has its history in the preceding understanding of
nature and mind, and their relationships. This history extends even as far as the pre-
Socratics, and I shall address some of these more distant historical connections later
in this study. However, the most significant historical trajectory of this study prior
to the birth of quantum theory, inaugurated by Planck’s discovery of the quantum of
action, h, begins with scientific modernity. There is no modernity other than
Preface ix
scientific, because modernity is defined, partially but decisively, by the rise of mod-
ern, mathematical-experimental, sciences of nature.
Consider John Milton’s description of chaos in Paradise Lost. This description
and Milton’s poem itself were written in the aftermath of the rise of mathematical-
experimental science, at that stage physics and astronomy, with Copernicus, Kepler,
and Galileo; and the poem was a response to a different world that emerged with and
because of this rise, a world to which we now refer as the world of modernity.
Milton’s stated aim in writing the poem is “to justify the ways of God to man,” with
“justify” referring to both the nature and the justness of these ways (Milton 2004,
p. 3, Paradise Lost, Book I, ll. 25–26). But why would one have needed such a
book? Don’t we already have the Bible that should do so? Well, not exactly, or
rather the Bible, Milton realized, was no longer sufficient to do so. As is clear from
Milton’s references in Paradise Lost to post-Copernican astronomy, and Galileo’s
and Boyle’s physics, Milton acutely realized that the world he lived in, the world of
modernity, was defined by, in Galileo’s words, “new mathematical sciences of
nature,” which brought mathematics and experiment together (Galilei 1991).
“Modern science,” M. Heidegger says, “is experimental because of its mathematical
project” (Heidegger 1967, p. 93). The world, as envisioned by Milton, was post-
Copernican and post-Galilean. R. Boyle has already conducted his famous experi-
ments on the properties of air and the existence of the vacuum, and Newton, Milton’s
equally famous fellow Cambridge graduate, was soon to appear on this stage and to
shape the thinking of modernity even more decisively. (Both Boyle and Newton had
major alchemical and theological interests, and left voluminous writings on these
subjects.) It was no longer the world of the Bible, and Milton reread or re- envisioned
the Bible as consistent with this new world. For Milton, God created the world as
understood by modern science and (they are, again, inseparable) scientific moder-
nity. This world called for a new justification of the ways of God to man, assuming
that this justification is possible, given that this new world compelled some to deny
this possibility, or the existence of God in the first place. The question of this justi-
fication or its possibility, which is still with us, is well outside the scope of this
study. But Milton’s argument for the extraordinary complexity of this world, which,
however it came about, requires the utmost reach of and may ultimately be beyond
human thought, is relevant to this project. This complexity and this relevance are
shown, for example and in particular, by Milton’s description of chaos in the poem:
Before their eyes in sudden view appear
The secrets of the hoary Deep—a dark
Illimitable Ocean without bound,
Without dimension: where length, breadth, and height,
And time, and place, are lost; where eldest Night
And Chaos, ancestors of Nature, hold
Eternal anarchy, amidst the noise
Of endless wars, and by confusion stand.
For Hot, Cold, Moist, and Dry, four champions fierce,
Strive here for maistrie, and to battle bring
Their embryon atoms: they around the flag
Of each his faction, in their several clans,
Light-armed or heavy, sharp, smooth, swift or slow,
x Preface
Swarm populous, unnumbered as the sands
Of Barca or Cyrene’s torrid soil,
Levied to side with warring winds, and poise
Their lighter wings. To whom these most adhere,
He rules a moment: Chaos umpire sits,
And by decision more embroils the fray
By which he reigns: next him, high arbiter,
Chance governs all. Into this wild Abyss,
The womb of Nature, and perhaps her grave,
Of neither Sea, nor Shore, not Air, nor Fire,
But all these in their pregnant causes mixed
Confus’dly, and which thus must ever fight,
Unless th’ Almighty Maker them ordain
His dark materials to create more worlds—
(Milton 2004, p. 20, Paradise Lost, Book II, 890–916)
The physical universe in this view is, thus, chaos, unless order emerges from it,
and this happens continuously, too, even if, generally, without giving this order sta-
bility. Milton’s description is presciently close to the understanding of the ultimate
constitution of nature arising from quantum theory, arguably more so than Lucretius’s
atomism in De Rerum Natura (Lucretius 2009), commonly claimed to be the main
precursor of modern atomic theory (along with Leucippus and Democritus, and then
Epicurus, on whose ideas Lucretius relies) and one of Milton’s sources. Boyle’s
experiments were undoubtedly on Milton’s mind as well. Milton’s conception does
not quite reach the radical form of this understanding to be advocated in this book.
Both randomness and chance, and the birth and disappearance of “particles” in
chaos, and thus unstable, fleeting nature of any order that might emerge in and from
it (unless some power manages to be stabilized and built on this order), are all part
of this book’s view of nature at the ultimate (quantum) level of its constitution. The
second aspect just mentioned is specifically found in high-energy regimes and
reflects or is reflected in the concept of virtual particle formation in quantum field
theory, according to which the unstable, fleeting forms of order emerge from and
disappear back into the foaming bubbling of chaos. This is what J. A. Wheeler refers
to as “quantum foam” (Wheeler and Ford 2000, pp. 245–263). However, according
to this study’s view, the ultimate character of this constitution, of Milton’s “embryon
atoms,” which we now refer to as “elementary particles” (still an unsettled concept
in fundamental physics, as is its companion concept, that of quantum field), is
“dark” beyond the reach of our understanding or possibly even any conception we
can form. This view is closer to, but still ultimately transcends, the ancient Greek
sense of chaos as areton or alogon, as that which is beyond all comprehension, than
to Milton’s conception of chaos here. On the other hand, Milton does appear to
imply that our ways of experiencing the world and conceptions we could form of it,
such as space, time, and causality (Kant’s three great a priori givens of our thought),
are “lost,” that is, no longer applicable to chaos. Milton was certainly aware of the
ancient Greek’s idea of chaos as areton or alogon. So perhaps he was closer to the
argument of this book on this point, except for the ultimately theological nature of
his thinking. This book is concerned with the u nrepresentable and possibly unthink-
Preface xi
able “dark materials” of nature as they appear in quantum physics, placed outside or
even assumed to be incompatible with theology. I prefer to leave theology to Milton.
If anything, this study’s understanding of the physical world, also because our inter-
action with it is governed by probabilistic thinking, is closer to the world of
Shakespeare’s plays (often invoked by Wheeler [e.g., Wheeler 1983, p. 204]), which
tend to put the theological aside. They leave it to us “to take arms against a sea of
troubles,” a sea, a place governed by chance and probability (Shakespeare 2005,
p. 700, Hamlet, III.2.55-87). The sea is often invoked by Shakespeare as such a
place, and Wheeler’s reference just mentioned is The Tempest (Shakespeare 2005,
p. 1238, Act IV.1, 148–158). As Nestor says in Troilus and Cressida:
… In the reproof of chance
Lies the true proof of men. The sea being smooth,
How many shallow bauble-boats dare sail
Upon her patient breast, making their way
With those of nobler bulk!
But let the ruffian Boreas once enrage
The gentle Thetis, and anon behold
The strong-ribbed bark through liquid mountains cut,
Bounding between the two most elements
Like Perseus’s horse. Where’s then the saucy boat
Whose weak untimbered sides but even now
Co-rivalled greatness? Either to harbor fled,
Or made a toast for Neptune. Even so
Doth valor’s show and valor’s worth divide
In storms of fortune. For in her ray and brightness
The herd has more annoyance by the breeze
Than by the tiger; but when the splitting wind
Makes flexible the knees of knotted oaks,
And flies fled under shade, why, then the thing of courage,
As rous’d with rage, with rage does sympathize,
And with an accent tun’d in selfsame key
Retorts to chiding fortune.
(Shakespeare 2005, p. 749, Act I.iii.33–54).
Shakespeare’s music is the music of the sea, the music of chance and its complex
harmonies, mixing chaos and order—chaosmic harmonies, as they were called by
James Joyce, from whose Finnegans Wake M. Gell-Mann famously borrowed the
term “quark” (Joyce 2012, p. 118). These chaosmic harmonies are opposed to the
music of the spheres, that of Pythagoras or that of Kepler, another contemporary of
Shakespeare. As my epigraph suggests, however, Joyce’s masterpiece was in turn
influenced by quantum theory, not inconceivably by the discovery of antimatter,
which was widely discussed at the time, just as the Higgs boson or black holes are
now, and was known to Joyce (Joyce 2012, pp. 383, 149). In Joyce’s novel words
transform into each other just as particles do in high-energy quantum physics. The
appearance of Thetis in Shakespeare’s passage is not by chance, and she is mentioned,
again, in the play: Thetis is the mother of Achilles, the greatest of heroes. It is the
rage of Achilles and his concern for the lack of virtue where The Iliad of Homer
begins. While the chance to kill Achilles is small, it is bound to happen at some point