Quantum field theory: spin zero

226 Pages·2005·1.011 MB·English

by  Srednicki M.

Quantum Field Theory
Part I: Spin Zero
Mark Srednicki
Department of Physics
University of California
Santa Barbara, CA 93106
[email protected]
This is a draft version of Part I of a three-part introductory textbook on
quantum field theory.
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Part I: Spin Zero
0) Preface
1) Attempts at Relativistic Quantum Mechanics (prerequisite: none)
2) Lorentz Invariance (prerequisite: 1)
3) Relativistic Quantum Fields and Canonical Quantization (2)
4) The Spin-Statistics Theorem (3)
5) The LSZ Reduction Formula (3)
6) Path Integrals in Quantum Mechanics (none)
7) The Path Integral for the Harmonic Oscillator (6)
8) The Path Integral for Free Field Theory (3, 7)
9) The Path Integral for Interacting Field Theory (8)
10) Scattering Amplitudes and the Feynman Rules (5, 9)
11) Cross Sections and Decay Rates (10)
12) The Lehmann-K¨all´en Form of the Exact Propagator (9)
13) Dimensional Analysis with h¯ = c = 1 (3)
14) Loop Corrections to the Propagator (10, 12, 13)
15) The One-Loop Correction in Lehmann-K¨all´en Form (14)
16) Loop Corrections to the Vertex (14)
17) Other 1PI Vertices (16)
18) Higher-Order Corrections and Renormalizability (17)
19) Perturbation Theory to All Orders: the Skeleton Expansion (18)
20) Two-Particle Elastic Scattering at One Loop (10, 19)
21) The Quantum Action (19)
22) Continuous Symmetries and Conserved Currents (8)
23) Discrete Symmetries: P, C, T, and Z (22)
24) Unstable Particles and Resonances (10, 14)
25) Infrared Divergences (20)
26) Other Renormalization Schemes (25)
27) Formal Development of the Renormalization Group (26)
28) Nonrenormalizable Theories and Effective Field Theory (26)
29) Spontaneous Symmetry Breaking (3)
30) Spontaneous Symmetry Breaking and Loop Corrections (19, 29)
31) Spontaneous Breakdown of Continuous Symmetries (29)
32) Nonabelian Symmetries (31)
2
Quantum Field Theory Mark Srednicki
0: Preface
This is a draft version of Part I of a three-part introductory textbook
on quantum field theory. The book is based on a one-year course that I
have taught off-and-on for the past 20 years. My goal is to present the basic
concepts and formalism of QFT as simply and straightforwardly as possible,
emphasizingitslogicalstructure. Tothisend,Ihavetriedtotakethesimplest
possible example of each phenomenon, and then to work through it in great
detail.
In this part, Spin Zero, the primary example is ϕ3 theory in six space-
time dimensions. While this theory has no apparent relevance to the real
world, it has many advantages as a pedagogical case study. For example,
unlike ϕ4 theory, ϕ3 theory has nontrivial renormalization of the propagator
at one loop. In six dimensions, the ϕ3 coupling is dimensionless, and asymp-
totically free. The theory is simple enough to allow a complete calculation
of the one-loop ϕϕ ϕϕ scattering amplitude, renormalized in an on-shell
→
scheme. This amplitude is singular in the limit of zero particle mass, and
this leads inexorably to an examination of infrared divergences, alternative
renormalization schemes, and the renormalization group.
I have tried to make this book user friendly, both for students and in-
structors. Each of the three main parts is divided into numerous sections;
each section treats a single topic or idea, and is as self-contained as possible.
An equation is rarely referenced outside of the section in which it appears;
if an earlier formula is needed, it is repeated. The book’s table of contents
includes, for each section, a list of the sections that serve as the immedi-
ate prerequisites. This allows instructors to reorder the presentation to fit
individual preference, and students to access topics of interest quickly.
Eventually, there will be more problems, and solutions to most of them.
Any comments, suggestions, corections, etc, would be very welcome.
Please send them to [email protected].
Please note: there are no references. For thorough guides to the litera-
ture, see, for example, Weinberg’s The Quantum Theory of Fields, Peskin &
Schroeder’s Introduction to Quantum Field Theory, or Siegel’s Fields.
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Quantum Field Theory Mark Srednicki
1: Attempts at relativistic quantum mechanics
In order to combine quantum mechanics and relativity, we must first
understand what we mean by “quantum mechanics” and “relativity”. Let us
begin with quantum mechanics.
Somewhere in most textbooks on the subject, one can find a list of the
“axioms of quantum mechanics”. These include statements along the lines
of
The state of the system is represented by a vector in Hilbert space.
Observables are represented by hermitian operators.
The measurement of an observable always yields one of its eigenvalues as
the result.
And so on. We do not need to review these closely here. The axiom we
need to focus on is the one that says that the time evolution of the state of
the system is governed by the Schr¨odinger equation,
∂
ih¯ ψ,t = H ψ,t , (1)
∂t| i | i
where H is the hamiltonian operator, representing the total energy.
Let us consider a very simple system: a spinless, nonrelativistic particle
with no forces acting on it. In this case, the hamiltonian is
1
H = P2 , (2)
2m
where m is the particle’s mass, and P is the momentum operator. In the
position basis, eq.(1) becomes
∂ h¯2
ih¯ ψ(x,t) = 2ψ(x,t) , (3)
∂t −2m∇
where ψ(x,t) = x ψ,t is the position-space wave function. We would like
h | i
to generalize this to relativistic motion.
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The obvious way to proceed is to take
H = + P2c2 +m2c4 , (4)
q
which gives the correct energy-momentum relation. If we formally expand
this hamiltonian in inverse powers of the speed of light c, we get
1
H = mc2 + P2 +... . (5)
2m
This is simply a constant (the rest energy), plus the usual nonrelativistic
hamiltonian, eq.(2), plus higher-order corrections. With the hamiltonian
given by eq.(4), the Schr¨odinger equation becomes
∂
ih¯ ψ(x,t) = + h¯2c2 2 +m2c4 ψ(x,t) . (6)
∂t − ∇
q
Unfortunately, this equation presents us with a number of difficulties. One
is that it apparently treats space and time on a different footing: the time
derivative appears only on the left, outside the square root, and the space
derivatives appear only on the right, under the square root. This asymmetry
between space and time is not what we would expect of a relativistic theory.
Furthermore, if we expand the square root in powers of 2, we get an infinite
∇
number of spatial derivatives acting on ψ(x,t); this implies that eq.(6) is not
local in space.
We can alleviate these problems by squaring the differential operators on
each side of eq.(6) before applying them to the wave function. Then we get
∂2
h¯2 ψ(x,t) = h¯2c2 2 +m2c4 ψ(x,t) . (7)
− ∂t2 − ∇
(cid:16) (cid:17)
This is the Klein-Gordon equation, and it looks a lot nicer than eq.(6). It
is second-order in both space and time derivatives, and they appear in a
symmetric fashion.
To better understand the Klein-Gordon equation, let us consider in more
detail what we mean by “relativity”. Special relativity tells us that physics
looks the same in all inertial frames. To explain what this means, we first
supposethatacertainspacetimecoordinatesystem(ct,x)represents(byfiat)
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an inertial frame. Let us define x0 = ct, and write xµ, where µ = 0,1,2,3,
in place of (ct,x). It is also convenient (for reasons not at all obvious at
this point) to define x = x0 and x = xi, where i = 1,2,3. This can be
0 i
−
expressed more elegantly if we first introduce the metric of flat spacetime,
1
−
+1
g =  . (8)
µν +1
 
 +1
 
 
We then have x = g xν, where a repeated index is summed.
µ µν
To invert this formula, we introduce the inverse of g, which is confusingly
also called g, except with both indices up:
1
−
+1
gµν =  . (9)
+1
 
 +1
 
 
We then have gµνg = δµ , where δµ is the Kronecker delta (equal to one
νρ ρ ρ
if its two indices take on the same value, zero otherwise). Now we can also
write xµ = gµνx .
ν
It is a general rule that any pair of repeated (and therefore summed) in-
dices must consist of one superscript and one subscript; these indices are said
to be contracted. Also, any unrepeated (and therefore unsummed) indices
must match (in both name and height) on the left- and right-hand sides of
any valid equation.
Now we are ready to specify what we mean by an inertial frame. If the
coordinates xµ represent an inertial frame (which they do, by assumption),
then so do any other coordinates x¯µ that are related by
x¯µ = Λµ xν +aµ , (10)
ν
where Λµ is a Lorentz transformation matrix and aµ is a translation vector.
ν
Both Λµ and aµ are constant (that is, independent of xµ). Furthermore,
ν
Λµ must obey
ν
g Λµ Λν = g . (11)
µν ρ σ ρσ
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Eq.(11) ensures that the invariant squared distance between two different
spacetime points that are labeled by xµ and xµ in one inertial frame, and by
′
x¯µ and x¯µ in another, is the same. This squared distance is defined to be
′
(x x)2 g (x x)µ(x x)ν = (x x)2 c2(t t)2 . (12)
′ µν ′ ′ ′ ′
− ≡ − − − − −
In the second frame, we have
(x¯ x¯)2 = g (x¯ x¯)µ(x¯ x¯)ν
′ µν ′ ′
− − −
= g Λµ Λν (x x)ρ(x x)σ
µν ρ σ ′ ′
− −
= g (x x)ρ(x x)σ
ρσ ′ ′
− −
= (x x)2 , (13)
′
−
as desired.
When we say that physics looks the same, we mean that two observers
(Alice and Bob, say) using two different sets of coordinates (representing two
different inertial frames) should agree on the predicted results of all possible
experiments. In the case of quantum mechanics, this requires Alice and Bob
to agree on the value of the wave function at a particular spacetime point,
a point which is called x by Alice and x¯ by Bob. Thus if Alice’s predicted
¯ ¯
wave function is ψ(x), and Bob’s is ψ(x¯), then we should have ψ(x) = ψ(x¯).
¯
Furthermore, in order to maintain ψ(x) = ψ(x¯) throughout spacetime, ψ(x)
¯
and ψ(x¯) should obey the identical equations of motion. Thus a candidate
wave equation should take the same form in any inertial frame.
Let us see if this is true of the Klein-Gordon equation, eq.(7). We first
introduce some useful notation for spacetime derivatives:
∂ 1 ∂
∂ = + , , (14)
µ ≡ ∂xµ c∂t ∇!
∂ 1 ∂
∂µ = , . (15)
≡ ∂xµ −c∂t ∇!
Note that
∂µxν = gµν , (16)
so that our matching-index-height rule is satisfied.
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¯
If x¯ and x are related by eq.(10), then ∂ and ∂ are related by
∂¯µ = Λµ ∂ν . (17)
ν
To check this, we note that
∂¯ρx¯σ = (Λρ ∂µ)(Λσ xν +aµ) = Λρ Λσ (∂µxν) = Λρ Λσ gµν = gρσ , (18)
µ ν µ ν µ ν
as expected. The last equality in eq.(18) is another form of eq.(11); see
section 2.
We can now write eq.(7) as
h¯2c2∂2ψ(x) = ( h¯2c2 2 +m2c4)ψ(x) . (19)
− 0 − ∇
After rearranging and identifying ∂2 ∂µ∂ = ∂2 + 2, we have
≡ µ − 0 ∇
( ∂2 +m2c2/h¯2)ψ(x) = 0 . (20)
−
This is Alice’s form of the equation. Bob would write
( ∂¯2 +m2c2/h¯2)ψ¯(x¯) = 0 . (21)
−
Is Bob’s equation equivalent to Alice’s equation? To see that it is, we set
¯
ψ(x¯) = ψ(x), and note that
∂¯2 = g ∂¯µ∂¯ν = g Λµ Λµ ∂ρ∂σ = ∂2 . (22)
µν µν ρ σ
Thus, eq.(21) is indeed equivalent to eq.(20). The Klein-Gordon equation
is therefore manifestly consistent with relativity: it takes the same form in
every inertial frame.
This is the good news. The bad news is that the Klein-Gordon equation
violates one of the axioms of quantum mechanics: eq.(1), the Schr¨odinger
equation in its abstract form. The abstract Schr¨odinger equation has the
fundamental property of being first order in the time derivative, whereas the
Klein-Gordon equation is second order. This may not seem too important,
but in fact it has drastic consequences. One of these is that the norm of a
state,
ψ,t ψ,t = d3x ψ,t x x ψ,t = d3xψ (x)ψ(x), (23)
∗
h | i h | ih | i
Z Z
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is not in general time independent. Thus probability is not conserved. The
Klein-Gordon equation obeys relativity, but not quantum mechanics.
Dirac attempted to solve this problem (for spin-one-half particles) by
introducing an extra discrete label on the wave function, to account for spin:
ψ (x), a = 1,2. He then tried a Schr¨odinger equation of the form
a
∂
ih¯ ψ (x) = ih¯c(αj) ∂ +mc2(β) ψ (x) , (24)
a ab j ab b
∂t −
(cid:16) (cid:17)
where all repeated indices are summed, and αj and β are matrices in spin-
space. This equation, the Dirac equation, is consistent with the abstract
Schr¨odinger equation. The state ψ,a,t carries a spin labela, andthe hamil-
| i
tonian is
H = cP (αj) +mc2(β) , (25)
ab j ab ab
where P is a component of the momentum operator.
j
SincetheDiracequationislinearinbothtimeandspacederivatives, ithas
a chance to be consistent with relativity. Note that squaring the hamiltonian
yields
(H2) = c2P P (αjαk) +mc3P (αjβ +βαj) +(mc2)2(β2) . (26)
ab j k ab j ab ab
Since P P is symmetric on exchange of j and k, we can replace αjαk by its
j k
symmetric part, 1 αj,αk , where A,B = AB+BA isthe anticommutator.
2{ } { }
Then, if we choose matrices such that
αj,αk = 2δjkδ , αj,β = 0 , (β2) = δ , (27)
ab ab ab ab ab
{ } { }
we will get
(H2) = (P2c2 +m2c4)δ . (28)
ab ab
Thus, the eigenstates of H2 are momentum eigenstates, with H2 eigenvalue
p2c2 + m2c4. This is, of course, the correct relativistic energy-momentum
relation. While it is outside the scope of this section to demonstrate it, it
turns out that the Dirac equation is fully consistent with relativity provided
the Dirac matrices obey eq.(27). So we have apparently succeeded in con-
structing a quantum mechanical, relativistic theory!
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There are, however, some problems. First, the Dirac matrices must be at
least 4 4, and not 2 2 as we would like (in order to account for electron
× ×
spin). Tosee this, notethatthe 2 2Pauli matricesobey σi,σj = 2δij, and
× { }
are thus candidates for the Dirac αi matrices. However, there is no fourth
matrix that anticommutes with these three (easily proven by writing down
the most general 2 2 matrix and working out the three anticommutators
×
explicitly). Also, the Dirac matrices must be even dimensional. To see this,
first define the matrix γ βα1α2α3. This matrix obeys γ2 = 1 and also
≡
γ,αi = γ,β = 0. Hence, using the cyclic property of matrix traces on
{ } { }
γβγ, we have Trγβγ = Trγ2β = Trβ. On the other hand, using βγ = γβ,
−
we also have Trγβγ = Trγ2β = Trβ. Thus, Trβ is equal to minus itself,
− −
and hence must be zero. (Similarly, we can show Trαi = 0.) Also, β2 = 1
implies that the eigenvalues of β are all 1. Because β has zero trace, these
±
eigenvalues must sum to zero, and hence the dimension of the matrix must
be even. Thus the Dirac matrices must be at least 4 4, and it remains for
×
us to interpret the two extra possible “spin” states.
However, these extra states cause a more severe problem than a mere
overcounting. Acting on a momentum eigenstate, H becomes the matrix
cα p + mc2β. The trace of this matrix is zero. Thus the four eigenvalues
·
must be +E(p),+E(p), E(p), E(p), where E(p) = +(p2c2 + m2c4)1/2.
− −
The negative eigenvalues are the problem: they indicate that there is no
ground state. In a more elaborate theory that included interactions with
photons, thereseems tobenoreasonwhy apositiveenergy electroncouldnot
emit a photon and drop down into a negative energy state. This downward
cascade could continue forever. (The same problem also arises in attempts
to interpret the Klein-Gordon equation as a modified form of quantum me-
chanics.)
Dirac made a wildly brilliant attempt to fix this problem of negative
energy states. His solution is based on an empirical fact about electrons:
they obey the Pauli exclusion principle. It is impossible to put more than
one of them in the same quantum state. What if, Dirac speculated, all the
negative energy states were already occupied? In this case, a positive energy
electron could not drop into one of these states, by Pauli exclusion!
Manyquestions immediately arise. Whydon’t weseethenegativeelectric
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