The Big Picture
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theory can be defined by saying what the action for the system is, and then looking for motions that minimize it.
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In quantum mechanics the action appears again, but with a twist. Feynman put for-
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ward an approach in which we can think of a quantum system as taking every path, not just 11
the classically allowed one. To each path we associate a certain phase factor, exp{ iS}. This 12
notation tells us to take a constant called Euler’s number, e = 2.7181 . . . and raise it to the power of i, the imaginary number given by the square root of –1, times the action S for 13
the path.
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The phase factor exp{ iS} is a complex number, with both a real part and an imaginary 15
part. Each will sometimes be positive and sometimes be negative. Summing up all the
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contributions for all the paths will generally involve a bunch of positive numbers and a bunch of negative numbers, and everything will cancel out, or nearly so, leaving us with a 17
small answer. The exception is when a group of nearby paths have very similar values for 18
the action; then their phase factors will be similar, and adding them up will accumulate 19
rather than canceling out. This happens exactly when the action is near a minimum value, which corresponds to the classically allowed path. So the largest quantum probability gets 20
associated with evolution that looks almost classical. That’s why our everyday world is well 21
modeled by classical mechanics; it’s classical behavior that gives the largest contributions 22
to the probability of quantum transitions.
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We can take our equation apart, piece by piece.
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Look at the part of the equation labeled “quantum mechanics.” That’s where the am-
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plitude is being written as an integral (the ∫ symbol) over a collection of fields, followed by 27
“exp i . . . ” The fields that we’re including are indicated in the notation [ Dg][ DA][ Dψ][ DΦ].
The letter D just means “here are the infinitesimal quantities we’re going to add up in our 28
integral,” and the other symbols stand for the fields themselves. The gravitational field is g; 29
the other bosonic force fields (electromagnetism, strong and weak nuclear forces) are
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grouped under A; all of the fermions are collectively labeled ψ (Greek letter psi); and the 31
Higgs boson is Φ (Greek letter phi). The notation “exp” means “e to the power . . .”; i is the square root of –1; and everything following i is the action S for the Core Theory. So quan-32
tum mechanics enters our expression by saying, “Integrate, over all of the paths that all of 33
the fields can take, a quantity given by raising e to the power of i times the action.”
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The action itself is where all of the fun is happening. Many professional particle phys-35S
icists spend a good fraction of their lives writing down different possible actions for different collections of fields. But everyone starts with this one, for the Core Theory.
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The action is an integral over all of space, and over the time period in between the
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initial configuration and the final configuration. That’s what the notation ∫ d 4 x is doing; 02
x stands for the coordinates on all the dimensions of spacetime, and the 4 is reminding us that spacetime is four- dimensional. There’s an extra factor lurking under the “spacetime”
03
label, which is the square root of something called – g. As you might guess from the letter 04
g, this has something to do with gravity, and in particular the fact that spacetime is 05
curved; this piece accounts for the fact that the volume of spacetime (over which we are 06
integrating) is affected by how spacetime is curved.
All of the terms inside the square brackets [] are the different contributions to the
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action from all of our various fields: both their intrinsic properties and how they are in-08
teracting with one another. They fall into the categories of “gravity,” “other forces,” “mat-09
ter,” and “Higgs.”
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The “gravity” term is pretty simple; that reflects the pristine elegance of Einstein’s theory of general relativity. The quantity R is called the curvature scalar, and characterizes 11
how much of a certain kind of spacetime curvature is present at any one point. It’s multi-12
plied by a constant, m 2/ 2, where m is the Planck mass. That’s just a funny way of express-p
p
13
ing Newton’s gravitational constant G, which characterizes the strength of gravity: m 2 =
p
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1/(8 πG). I’m using “natural units,” in which both the speed of light and Planck’s constant of quantum mechanics are set equal to unity. The curvature scalar R can be calculated 15
from the gravitational field, and the action for general relativity is simply proportional to 16
the integral of R over a region of spacetime. Minimizing that integral gives you Einstein’s 17
field equation for gravity.
Next up, we have the term labeled “other forces,” which includes two appearances of
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a quantity F and a few superscripts and subscripts. F is called the field strength tensor, and 19
in our notation it includes contributions from electromagnetism and the strong and weak 20
nuclear forces. Essentially the field strength tensor tells us how much the field is twisting 21
and vibrating through spacetime, much as the curvature scalar tells us how much the geometry of spacetime itself is twisting and vibrating. For electromagnetism, the field
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strength tensor incorporates both the electric and the magnetic field.
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Here, and elsewhere in the equation, those superscripts and subscripts label different 24
subquantities, such as which field we’re talking about (photon, gluon, W or Z boson), but 25
also which part of the field, such as “the part of the electric field pointing along the x-axis.”
When you see two quantities (like the two F’s in this term) with the same indices on them, 26
that’s code for “sum over all of the possibilities.” This is a very compact notation, allowing 27
us to hide great complexity in just a few symbols; that’s why this one term encompasses 28
the contributions from all the different force fields.
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Things get a bit more complicated when we look at the part of our equation labeled “mat-31
ter.” The matter fields are fermions, represented collectively by the letter ψ. As with the 32
bosons, this one symbol includes all of the fermions at once. The first term has two appear-33
ances of ψ, one of the Greek letter γ (gamma), and another D. That γ stands for the Dirac matrices, introduced by British physicist Paul Dirac; they play an essential role in how 34
fermions behave, including the fact that fermions generally have antiparticles as well as S35
particles. The D in this case stands for a derivative, or rate of change, of the field. So this N36
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>
term is doing the same thing for the fermions that the previous terms did for the force bosons: it tells us how much the fields are changing through space and time. But there is 02
something hidden in that derivative (the magic of compact notation, again): a coupling, or 03
interaction, between the fermions and the force bosons, which depends on how the fer-
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mions are charged. The way that an electron interacts with a photon, for example, is char-05
acterized by this term in the action.
The term next to it involves a different kind of coupling, between the fermions and the 06
Higgs field Φ. Unlike the rest of the action of the Core Theory, the interaction between 07
the Higgs and fermions is somewhat baroque and unappealing. But there it is: two ψ’s and 08
one Φ, telling us that this term encapsulates how fermions and the Higgs field interact with each other. Two things make it complicated. One is that symbol
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V, known as a mix-
ij
ing matrix. It keeps track of the fact that fermions can “mix” with one another— when a 10
top quark decays, for example, it actually decays into a particular mixture of down,
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strange, and bottom quarks.
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The other complication is that you see that one fermion field has a subscript L and the other a subscript R. These stand for “ left- handed” and “ right- handed” fields. Think 13
of lining up the thumb of your left hand along the direction of motion of a spinning
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particle. Your other fingers define a possible direction that the spin could be in; if that’s 15
what the particle is doing, it’s left- handed, while if it’s spinning the other way, it’s right-16
handed. The appearances of these subscripts in this term in the Core Theory is an indication that the theory treats left- handedness differently from right- handedness, at least at the 17
subatomic level. That’s a remarkable feature, but a necessary one, since nature treats left-18
handed and right- handed particles differently. That phenomenon, parity violation, came 19
as a shock to particle physicists when it was first discovered, but is now seen as simply 20
the sort of thing that can happen when you get these kinds of fields interacting with one another.
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The last bit of this term, “h.c.,” stands for hermitian conjugate. It’s a fancy way of saying 22
that the first term is a complex number, but the action needs to be a real number, so we’re 23
going to subtract out the imaginary part and be left with a purely real quantity.
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Finally we have the part of the action that is devoted to the Higgs field Φ. It’s pretty simple; the first part is the “kinetic” term, representing how much the field is changing, 25
and the second is the “potential” term, representing how much energy is locked up in the 26
field even when it’s not changing at all. It’s that second term that makes the Higgs field 27
special. Like any other field, it wants to be sitting peacefully with the lowest energy it can 28
have; unlike the other known fields, in its minimum- energy state the Higgs field itself does not vanish but has a nonzero value. Its potential energy is higher when the field is zero 29
than it is when the field is not zero. That’s what gives the Higgs field a presence even in 30
“empty space,” and allows it to affect all the other particles that move through it.
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So there we have it: the Core Theory in a nutshell. One equation that tells us the quantum 33
amplitude for the complete set of fields to go from some starting configuration (part of a 34
superposition inside a wave function) to some final configuration.
35S
We know that the Core Theory, and therefore this equation, can’t be the final story.
There is dark matter in the universe, which doesn’t fit comfortably into any of the known 36N
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fields. Neutrinos have mass, which can be accommodated by the equation we wrote down,
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but we haven’t experimentally verified that the terms we wrote are actually the ones that 02
are responsible for neutrino masses. Moreover, almost every physicist believes there are more particles and fields to be found, at higher masses and energies— but they must be 03
ones that either interact with us very weakly (like dark matter) or decay away very quickly.
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The Core Theory doesn’t even provide a complete theory of the fields that we know are
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there. That’s the problem, for example, with quantum gravity. The equation we wrote is 06
okay if the gravitational field is very weak, but it doesn’t work when gravity becomes strong, such as near the Big Bang or inside a black hole.
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That’s okay. Indeed, the theory’s limitations are built into the formalism. There is one 08
piece of notation in our equation that we haven’t yet mentioned: in the very first integral 09
sign, indicating that we’ll be summing over all the different field configurations through 10
time, there is a subscript reading k < Λ. Here k is the wave number of a particular mode of a field, and Λ is called the ultraviolet cutoff. Remember the viewpoint that Ken Wilson 11
advocated, as we discussed in chapter 24: we can think of every field as a combination of 12
modes, each constituting a vibration with a specific wavelength. The wave number is a way 13
of labeling these modes; larger k corresponds to smaller wavelength, and therefore higher 14
energies. So this notation is limiting the field configurations we include in the path integral to those that “don’t vibrate too energetically.” That means low- energy, weak- field 15
situations— but still enough to include all of the buzzing and bouncing of the particles 16
and fields that describe the world you see around you every day.
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The Core Theory, in other words, is an effective field theory. It has a very specific, well-defined regime of applicability— particles interacting with energies well below the ultra-18
violet cutoff Λ— and we don’t pretend that it’s accurate past that. It can describe the 19
gravitational pull of the sun on the Earth, but not what was happening at the Big Bang.
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There’s a lot going on here, material usually relegated to graduate- level physics courses.
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This condensed presentation can’t reasonably be expected to convey much understanding
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to anyone who isn’t already pretty familiar with the concepts.
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But it’s useful to see how the Core Theory underlying our everyday lives is extremely
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precise, rigid, and well defined. There is no ambiguity in it, no room to introduce important new aspects that we simply haven’t noticed yet.
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As science continues to learn more about the universe, we will keep adding to it, and
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perhaps we will even find a more comprehensive theory underlying it that doesn’t refer to 28
quantum field theory at all. But none of that will change the fact that the Core Theory is 29
an accurate description of nature in its claimed domain. The fact that we have successfully put together such a theory is one of the greatest triumphs of human intellectual history.
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References
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In this section, references are provided for quotations and the sources of various specific 14
topics in the text. When I thought it might not be obvious, the topic is defined by a word 15
or phrase immediately preceding the reference. The list is organized in chapter sequence but not all chapters have references.
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3. the World Moves by Itself
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History of momentum: Freely, J. (2010). Aladdin’s Lamp: How Greek Science Came to
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Europe through the Islamic World. Vintage Books.
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5. Reasons Why
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Just World Fallacy: Lerner, M. J., and C. H. Simmons. (1966). “Observer’s Reaction to the 22
‘Innocent Victim’: Compassion or Rejection?” Journal of Personality and Social
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Psychology 4 (2): 203.
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8. Memories and Causes
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Russell quote: Russell, B. (1913). “On the Notion of Cause.” Proceedings of the Aristotelian Society 13: 1– 26.
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14. Planets of belief
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Dorothy Martin: Tavris, C., and E. Aronson. (2006). Mistakes Were Made (But Not by 30
Me): Why We Justify Foolish Beliefs, Bad Decisions, and Hurtful Acts. Houghton Mifflin Harcourt.
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15. Accepting uncertainty
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Catechism of the Catholic Church: “Catechism of the Catholic Church— The Transmis-
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sion of Divine Revelation.” Accessed December 10, 2015. http:// www.vatican.va/
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archive/ ccc_ css/ archive/ catechism/ p1s1c2a2.htm.
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16. What Can We Know about the World without Actually looking at It?
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