The propagator and diffeomorphisms of an interacting field theory

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The propagator and diffeomorphisms of an interacting field theory

Master thesis

submitted to the

Institut f¨ ur Physik

Mathematisch-Naturwissenschaftliche Fakult¨at Humboldt-Universtit¨at zu Berlin

by

Paul-Hermann Balduf

in partial fulfillment of the requirements for the degree

Master of Science.

Adlershof, July 5, 2018

Referees:

Prof. Dr. Dirk Kreimer (supervisor)

Dr. Christian Bogner

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We consider a scalar quantum field theory where the fieldφis replaced by a diffeomor- phismφ=ρ+a1ρ²+a2ρ³+. . .. The fieldρconstitutes a modified quantum field theory defined implicitly by the diffeomorphism coefficients{a_j}_j and the Lagrange density of the underlying fieldφ. For a generic diffeomorphism, ρ is a non-renormalizable quantum field with infinitely many interaction vertices, even if φitself is a free field. In the case that φ is an interacting field itself, ρ obtains additional vertices proportional to the coupling parameters in the Lagrangian density ofφ.

We examine the general Lagrangian density of a scalar field with local interactions, L= 1

2∂_µφ∂^µφ−1

2m²φ²−

∞

X

s=3

λ_s s!φ^s,

and show that the S-matrix elements ofρcoincide with the ones of φ. This implies the fieldsφand ρare indistinguishable in experiments. In this sense diffeomorphisms form equivalence classes of interacting scalar quantum field theories.

On the other hand, n-point functions differ between φ and ρ if their momenta are offshell. Tuning the diffeomorphism coefficients{a_j}_j allows to change the behaviour of the offshelln-point functions ofρ. We use this freedom to eliminate all loop-corrections to the propagator ofρfor a fixed offshell four-momentump. Ifφhasφ^s-type interaction,

L= 1

2∂_µφ∂^µφ−1

2m²φ²− λ s!φ^s, and all tadpole graphs vanish, then the field formally given by

ρ(x) =φ(x)− λ

(s−1)!(p²−m²)φ^s−1(x) has a free propagator to all orders in perturbation theory, i.e.

ρ(p)ρ(−p)

=−i p²−m² .

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1 Introduction

1.1 Minkowski spacetime

Four-dimensional Minkowski spacetime M with speed of lightc= 1 is used throughout.

Four-vectors are underlined and three-vectors are bold such that the four-momentum is

p= p⁰, p¹, p², p³

= (E, p_x, p_y, p_z) = (E,p).

The metric is flat and has negative sign,

η =







1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1





 .

Four-dimensional spacetime indices are greek and summation is implicit,

p² =pµp^µ=ηµνp^µp^ν =E²−p². (1.1) Other indices are not summed over unless explicitly denoted. The spacetime derivative is∂_µ= _∂x^∂µ, especially the second derivative

∂µ∂^µ=∂0∂0−∂1∂1−∂2∂2−∂3∂3 =∂_t²− 4=.

Physical particles (i.e. “really existent” as opposed to virtual particles used in intermediate steps in computations) obey the relativistic energy-momentum-relation

E² =p²+m². (1.2)

Byeq. (1.1)the square of the four-momentum of a physical particle is its mass squared.

In four-dimensional momentum space, eq. (1.2) forms a surface called the mass shell.

Hence if a particle fulfillseq. (1.2)it is calledonshell, otherwise offshell.

Definition 1. For a four-momentump assigned to a particle with massm, the offshell parameter is defined as

xp=p²−m².

This parameter is zero if the momentum p belongs to a physical particle with mass m. The notation here follows slightly nontrivial rules: If the index is a letter like x_p, this letter indicates the name of the corresponding four-momentum. If the index is a number, it is the running number of some (usually canonically) numbered momenta like

x₁=p²

1−m². (1.3)

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If sums and differences appear in the index, these indicate the corresponding operations with momenta, not indices themselves. This is,

x1+2= p₁+p₂

2

−m², (1.4)

but generally x₁₊₂6=x₃. Note that x_p only depends on the magnitude of the momentum, for any four-momentump it is xp =x−p. The equation xp =xq does not imply p=q.

1.2 Fuss-Catalan numbers

Definition 2. For fixed a∈N0 and b∈N the Fuss-Catalan numbers are defined as A_m(a, b) = b

ma+b

ma+b m

=b (ma+b−1)!

(ma+b−m)!m!. Some useful values are

A₀(a, b) = 1 A_m(1,1) = 1 Am(1,2) =m Am(0, b) =

b m

.

Example 1: Fuss-Catalan numbers used in subsequent examples

The (ordinary) Catalan numbers{C_m}_m∈N={1,1,2,5,14,42,132,429, . . .} are Am(2,1) =Cm = 1

m+ 1 2m

m

= (2m)!

m!(m+ 1)!. (1.5)

In the context of φ⁴-theory we encounter the choice a= 3, b= 1, {A_m(3,1)}_m∈N=

1 2m+ 1

3m m

m∈N

={1,1,3,12,55,273,1428,7752. . .}.

Countless interpretations of the Fuss-Catalan numbers are known, see for example their entries A000108 (A_m(2,1)) or A001764 (A_m(3,1)) in the OEIS [Slo18]. Noteworthy, the Catalan numberCm is the number of different planar trees with n+ 1 leaves built of 3-valent vertices. Similarly,Am(s,1) counts the number of such trees if they are made ofs-valent vertices.

The generating function of the Fuss-Catalan numbers is defined by Ca,b(t) =

∞

X

m=0

t^mAm(a, b). (1.6)

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Lemma 1.1.

Iff for q∈R a functionR(t) obeys

R(t)−1 =tR^q(t) then forb∈R

R^b(t) =

∞

X

m=0

t^m b mq+b

mq+b m

.

Proof. This is [MSV06, theorem 2.1] with the replacement R(t) = 1 +w.

Bylemma 1.1the generating functioneq. (1.6)of the Fuss-Catalan-numbers is defined equivalently via its functional equation

Ca,b(t) = tC

a b

a,b(t) + 1b

. (1.7)

Fromeq. (1.6)and A₀(a, b) = 1 the boundary condition is

C_a,b(0) = 1. (1.8)

Forb= 1 anda= 2 or a= 3 the functional equation can be solved explicitly.

1.2.1 Generating function of A(2,1)

Seta= 2, b= 1, this gives the generating function of ordinary Catalan numbers defined implicitly viaeq. (1.7)

C2,1(t)≡C(t) =tC²(t) + 1. (1.9) Solving the quadratic equation, there are (for any given t) two solutions

C_1/2(t) = 1±√ 1−4t

2t . (1.10)

As can be seen infig. 1.1or computed explicitly, only one of them fulfills the boundary conditioneq. (1.8) C(0) = 1, consequently the generating function of Catalan numbers is

C(t) = 1−√ 1−4t

2t . (1.11)

As a generating function according to eq. (1.6), C(t) = C2,1(t) has to be real-valued.

This is the case for

t≤ 1 4.

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Figure 1.1:The two solutions Cj(t) fromeq.(1.10) of the functional equation (1.9). Solid real part, dashed imaginary part. To distinguish them, the imaginary parts have been slightly shifted off thex-axis. OnlyC₂(t)(red) fulfills the boundary conditionC(1) = 1 fromeq.(1.8).

1.2.2 Generating function of A(3,1)

Fora= 3, b= 1 set C(t)≡C_3,1(t), then eq. (1.7)becomes the cubic equation C³(t)−1

t

|{z}p

C(t) + 1 t

|{z}q

= 0. (1.12)

It can be solved with the Cardanic formulae [Car68]. Define

∆ =p 3

3

+q 2

2

= 27t−4 108t³ and two auxiliary functions

u(t) = ³ r

−q 2 +√

∆ = ³ s

−1 2t+

r−4 + 27t 108t³ , v(t) = ³

r

−q 2 −√

∆ = ³ s

−1 2t−

r−4 + 27t 108t³ .

The complex cube roots have to be chosen to guaranteeu·v =−^p₃ = _3t¹ as shown in fig. 1.2.

Introducing the cube roots of unity₁ = ⁻¹⁺ⁱ

√ 3

2 and ₂ = ⁻¹⁻ⁱ

√ 3

2 , the three solutions

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Figure 1.2: Functionsu(t)and v(t).

ofeq. (1.12) are

C1(t) =u+v

C₂(t) =₁u+₂v (1.13)

C₃(t) =₂u+₁v.

Figure 1.3:The three solutionsC_j(t)fromeq.(1.13). The solution fulfillingeq.(1.8),C(1) = 1, can be constructed joining C₁(t)andC₂(t).

To fulfill the boundary condition eq. (1.8), the correct generating function is C₁(t)

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fort <0 and and C(0) = 1 andC2(t) for t >0. Explicitly, this is

C_3,1(t) =







C₁(t) t <0

1 t= 0

C₂(t) t >0

(1.14)

=











3

r

−_2t¹ +

q27t−4 108t³ + ³

r

−_2t¹ −

q27t−4

108t³ t <0

−1+i√ 3 2

3

r

−_2t¹ +

q27t−4 108t³ −¹⁺ⁱ

√ 3 2

3

r

−_2t¹ −

q27t−4

108t³ t≥0.

The distinction of the two cases just means taking different cube roots. The generating functionC_3,1(t) has to be real-valued, this is the case only for

t≤ 4 27.

1.3 Bell polynomials

We follow the definitions of [KY17]. See also [FG05, chapter 7] for details and proofs of the properties listed below.

Definition 3. Fork∈N0, n∈N0 and k≤n, the Bell polynomials are Bn,k(x1, x2, x3, . . .) =X

S

n!

j₁!j₂!· · ·j_n! x1

1!

j1x2

2!

j2x3

3!

j3

· · ·xn

n!

jn

, where the sum runs over all {j_i}i∈{1,...,k} such that

S ={j_i ≥0 ∀i, j₁+j₂+j₃+. . .+j_k=k j₁+ 2j₂+ 3j₃+. . .+ (n−k)jn−k=n}.

Special values are

B0,0 = 1

B0,k = 0, k >0

B_n,0 = 0, n >0 (1.15)

B_n,k = 0, k > n Bn,1 =xn, n >0 Bn,n =xⁿ₁, n >0.

We will frequently need the generating function of Bell polynomials

∞

X

n=0 n

X

k=0

B_n,k(x₁, x₂, . . .)u^ktⁿ n!= exp



u

∞

X

j=1

x_jt^j j!



. (1.16)

Expanding the exponential function, exchanging summation order on the l.h.s. and extracting the coefficients ofu^k this becomes

∞

X

n=k

B_n,k(x₁, x₂, . . .)tⁿ n! = 1

k!





∞

X

j=1

x_jt^j j!





k

. (1.17)

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Bell polynomials have a striking combinatoric meaning: They count partitions of {1, . . . , n} intok nonempty disjoint subsets, i.e.

B_n,k(x₁, x₂, x₃, . . .) =X

P

x_|P₁_|· · ·x_|P_k_|, where

P ={∅ 6=Pi ⊆ {1, . . . , n} ∀i, Pi∩Pj =∅ ∀i6=j, P1∪. . .∪P_k ={1, . . . , n}}.

(1.18) A consequence of eq. (1.16) is that Bell Polynomials represent the coefficients of composed diffeomorphisms. Fa`a die Bruno’s Formula [Wei18a] for the n-th derivative

∂ⁿ_t of a composed function is

∂_tⁿf(g(t)) =

n

X

k=0

∂_t^kf

(g(t))·Bn,k ∂tg(t), ∂_t²g(t), ∂_t³g(t), . . . . In terms of power seriesf(t) =P∞

n=1f_n^t_n!ⁿ,g(t) =P∞

n=0g_n^t_n!ⁿ and f(g(t)) =:h(t) =

∞

X

n=0

hn

tⁿ n!

this becomes

h_n=

n

X

k=1

f_k·B_n.k(g₁, . . . , gn+1−k). (1.19)

1.4 Axiomatic quantum field theory

We consider scalar quantum fields exclusively, these are fields with spin zero. The following part follows Lutz Klaczynski’s PhD thesis [Kla16].

1.4.1 Wightman axioms

The Wightman axioms are a mathematically rigorous way of defining quantum fields.

They read

1. The states of the physical system are described by vectors in a separable Hilbert space hequipped with a strongly unitary representation (a,Λ)→ U(a,Λ) of the connected Poincar´e group P₊^↑ (i.e. the group of orthochronous proper Lorentz transformations and shifts). Moreover there is a unique vacuum Ψ₀∈hwhich is invariant under these Poincar´e transformations,U(a,Λ)Ψ0 = Ψ0.

2. The generator of the translation subgroup i ∂

∂a^µU(a,1)

a=0=p_µ

has its spectrum inside the closed forward light cone: σ(p)⊂V₊and the generator of time translations (the Hamiltonian) has nonnegative eigenvalues,H =p₀ ≥0.

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3. For every Schwartz functionf ∈ S(M) there are operatorsφ1(f), . . . , φn(f) (called quantum fields) and their adjointsφ^†₁(f), . . . , φ^†n(f) onhsuch that the polynomial algebra

A(M) =D

φ_j(f), φ^†_j(f) :j∈ {1, . . . , n}E

C

has a stable common dense domainD⊂h. This means A(M)D⊂D

U(a,Λ)D⊂D ∀(a,Λ)∈ P₊^↑. Further, the vacuum is cyclic forA(M), this means

Ψ₀ ∈D

D₀ :=A(M)Ψ₀ ⊆Dis dense inh.

Finally, the maps

f →

Ψ φj(f)Ψ⁰

are tempered distributions onS(M) for all Ψ,Ψ⁰ ∈Dandj ∈ {1, . . . , n}.

4. The quantum fields transform under the unitary representation of the Poincar´e group according to

U(a,Λ)φ_j(f)U^†(a,Λ) =φ_j({a,Λ}f) on the domainDwhere

({a,Λ}f)(x) =f Λ⁻¹(x−a) is the Poincar´e-transformed test function.

5. Forf, g∈ S(M) with mutually spacelike-separated support, this is f(x)g(y)6= 0⇒(x−y)² <0,

the commutator of quantum fields vanishes:

[φj(f), φ_l(g)] =φj(f)φ_l(g)−φ_l(g)φj(f) = 0 ∀j, l∈ {1, . . . , n}.

1.4.2 Wightman distributions

Wightman distributions are then-fold vacuum expectation values of quantum fields W_n(f₁, . . . , f_n) =hΨ₀ φ(f₁)· · ·φ(f_n)Ψ₀i.

They fulfill a set of axioms equivalent to the Wightman axioms for the quantum fields themselves in the sense that given a set of Wightman distributions, there exists a

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corresponding quantum field theory fulfilling the Wightman axioms (Wightman’s reconstruction theorem, [Kla16, Theorem 10.1]).

Especially, if φis a free field without interaction, the 2-point distribution (where Ω0

instead of Ψ0 is used to stress the fact this state is the vacuum of a free theory) is hΩ₀ φ(f)φ(h)Ω₀i=

Z d⁴p

(2π)⁴f˜^∗(p)2πθ(p⁰)δ(p²−m²)˜h(p), (1.20) where ˜f is the Fourier transform of f. If one takes the test functions f and h to have

“sufficiently” small support around some pointsx,ysuch that they can be approximated by delta distributions

f(z)→δ(z−x), h(z)→δ(z−y) (1.21) then their Fourier transformations become

f˜^∗(p)→e^−ipx ˜h(p)→e^ipy and one can use the notation

φ(x) :=φ(f)

f(y)≈δ(y−x)

φ(p) :=φ( ˜f) _˜

f(p)≈exp(ipy).

Under this condition eq. (1.20) reduces to the function [Zwi17, eq. (7)], [Kla16, eq.

(9.5)]

∆₊ x−y, m² :=

Ω₀ φ(x)φ(y)Ω₀

(1.22)

=

Z d⁴p

(2π)³e^−ip(x−y)θ(p⁰)δ(p²−m²)

= 1 2

Z d³p (2π)³

e^−ip(x−y) pp²+m².

This is a well-defined function for x 6= y but has a pole at x = y. This divergence indicates that the replacementeq. (1.21)is not allowed. The physical assumption that a quantum field be a well-defined operator for any sharp spacetime point x ∈ M is impossible to satisfy by non-singular quantum fields [Kla16, thm. 9.1].

1.4.3 Haag‘s theorem

Two results severely restrict the behaviour of Wightman functions. Note that the Wightman functionseqs. (1.20)and (1.22)belong to free quantum fields, it is generally hard to compute a Wightman function of an interacting quantum field.

A free 2-point function is the correlation function of a free quantum field according to eq. (1.20). Similarly, a free field has certain free n-point functions and by Wightman’s reconstruction theorem if all correlation functions coincide with the ones of a free field, the field itself is free of interaction.

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The first theorem states that having a free 2-point function for a quantum field is sufficient that all higher n-point functions also coincide with the ones of a free field.

Conversely, the this implies that no interacting quantum field can “by chance” have a free 2-point function and be otherwise interacting.

Theorem 1.1 Jost-Schroer theorem.

If m ≥ 0 and φ is a scalar quantum field in Minkowski spacetime with the two-point Wightman function of a free scalar field ∆₊(x−y, m²) according toeq. (1.22), thenφ is a free field with mass m.

Proof. [Kla16, thm. 11.6]. The case m >0 is the original theorem by Jost and Schroer [Jos61], the casem= 0 was shown by Pohlmeyer in [Poh69].

The Jost-Schroer theorem underlines the importance of understanding the 2-point function of any quantum field under consideration.

The second theorem asserts that a free quantum field cannot be turned into an interacting one by an unitary transformation.

Theorem 1.2 Haag’s theorem.

Let φand φ0 be two Hermitian scalar quantum fields of equall massm ≥0, the latter of which is a free field. Assume the sharp-time limitsφ(t, f) andφ0(t, f) exist and form an irreducible set in their respective Hilbert spaceshandh₀ for this specific timet= 0.

If there is an isomorphism V :h₀ →h(i.e. an unitary map) such that at t= 0 φ(0, f) =V φ₀(0, f)V⁻¹

then φis a free field with massm.

Proof. [Kla16, thm. 11.7]. Knowing the Jost-Schroer theorem, the proof reduces to showing thatφ andφ₀ have the same 2-point Wightman function.

1.5 Perturbative quantum field theory

1.5.1 Scattering theory in the interaction picture

The perturbative treatment, althought its compatibility with axiomatic quantum field theory in some points appears unsatisfactory, is highly successfull at predicting exer- perimental observations and can be found in any QFT textbook. We follow [Das10;

BD67] and also the overview given in [Kla16]

Definition 4. The time ordering operator ˆT is for time-dependent operatorsφ(x), φ y defined as

Tˆ

φ(x)φ(y)

=θ(x⁰−y⁰)φ(x)φ(y) +θ(y⁰−x⁰)φ(y)φ(x).

The operator with the latest time is arranged to the left position in the product. This readily generalizes to any product of operators.

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The Feynman propagator is the time-ordered 2-point correlation function of the free scalar field and evaluates to

∆_F(x−y, m²) :=D

Ω0 Tˆ[φ(x)φ(y)]Ω0

E

(1.23)

=

(∆+ x−y, m²

x⁰ > y⁰

∆₊ y−x, m²

x⁰ < y⁰

= lim

→0

Z d⁴p (2π)⁴

e^−ip(x−y) p²−m²+i. It is also the Green function of the Klein-Gordon operator acting onx,

∂_µ∂^µ+m²

∆F(x−y, m²) =−δ(x−y).

In the following, unless noted otherwise, the term n-point function will always denote time-ordered n-point vacuum expectation values like eq. (1.23) and not Wightman distributions like eq. (1.22).

In the Heisenberg picture, the field operators φ(x) = φ(t,x) carry the full time evolution of the physical system. Conversely, in the Schr¨odinger picture, the fields (like any operators) are constant in timeφ=φ(x) and it is the states which change in time.

The (time-independent) HamiltonianH is the generator of time translations, hence φ(t,x) =e^iHtφ(x)e^−iHt

relates Heisenberg- and Schr¨odinger pictures if they agree at a timet= 0. The Hamil- tonian can be split into an interaction part and the Hamiltonian of the free field theory, H=H_int+H₀. Then one define the interaction picture by

φ₀(t,x) =e^iH⁰^tφ(x)e^−iH⁰^t. (1.24) Here φ(x) is the time-independent Schr¨odinger picture field operator. The subscript φ₀(t,x) is motivated by the fact that this field evolves in time by a free Hamiltonian,

H0= Z

d³xH= Z

d³x 1

2π²(x) + 1

2∇φ(x)∇φ(x) + 1

2m²φ²(x)

.

Combining both above definitions yields a relationship between Heisenberg picture and interaction picture,

φ(t,x) =e^iHte^−iH⁰^tφ0(t,x)e^iH⁰^te^−iHt

=V^†(t)φ0(t,x)V(t) (1.25)

with a time evolution operator

V(t) =e^iH⁰^te^−iHt. (1.26)

Hint = H −H0 is the time-independent interaction part of the Hamiltonian in the Schr¨odinger picture. It is a polynomial in the fields φ(x), therefore in the interaction picture it becomes time-dependent as well,

H_int,i(t) =V^†(t)H_intV(t). (1.27)

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For an evolution from timet1 tot2 in the interaction picture define U(t2, t1) =V(t2)V^†(t1) =e^iH⁰^t²e^−iHt²e^iHt¹e^−iH⁰^t¹

= ˆT

exp

−i Z t2

t1

dτ Hint,i(τ)

The operator U(∞,−∞) =: S gives the time evolution from infinite past to infinite future and is called the S-matrix. This matrix encodes the probability for an initial state|iito evolve from the infinite past to infinite future and reach a final state |fi by probability for evolution i→f =hf Sii. (1.28) Eventually, the goal of quantum field theory is to reproduce and predict experimental observations. In high energy physics, apart from masses of bound states, these are pri- marily cross-sections in scattering experiments. The Lehmann-Symanzik-Zimmermann reduction formula [LSZ55] relates a scattering process withn(both incoming and out- going counted) particles to n-point correlations functions of the underlying quantum field theory. Hence the main objective is to compute these correlation functions.

The standard procedure to obtain perturbative approximations to n-point functions consists of three steps: First, if Ψ0 is the vacuum of H and Ω0 is the vacuum of the free Hamiltonian H0, then the time-ordered n-point functions of the Heisenberg- and interaction picture are related via the Gell-Mann-Low formula [GL51]

hΨ0 T[φ(x₁)· · ·φ(x_n)]Ψ0i= hΩ0 T[Sφ0(x₁)· · ·φ0(x_n)]Ω0i

hΩ₀ SΩ₀i . (1.29)

Next, theS-matrix is expressed via Dyson’s series S = 1 +

∞

X

n=1

(−i)ⁿ n!

Z

· · · Z

d⁴x₁· · ·d⁴x_nT[H_int(x₁)· · ·H_int(x_n)] (1.30) which reduces the n-point function of an interacting theory to an infinite sum of correlation functions of the free theory. Finally, Wick’s theorem [Wic50] is used to reduce a n-point function of the free theory to all possible products of 2-point functions, i.e.

Feynman propagatorseq. (1.23), of the free theory.

Graphically, this can be depicted as a number of points connected with edges (which represent the propagators) such that the sum over all possible products equals the sum over all graphs fulfilling certain restrictions. These graphs are calledFeynman graphs and the Wick theorem reads

DΩ₀ Tˆ[:φⁿ₀¹(x₁) :· · ·:φⁿ₀^k(x_k) :]Ω₀E

=X

Γ∈G

c(Γ) Y

e∈E_Γ

i∆_F(x_start(e)−x_end(e), m²).

(1.31) Here, :. . .: indicates normal ordering andc(Γ) involves several factors to be discussed insection 1.5.2.

A detailled analysis shows that large classes of Feynman graphs can be ignored right away. Either they cancel against the denominator ofeq. (1.29)or they can be recovered trivially as products of smaller graphs. Eventually it is sufficient to calculate Graphs which are 1PI according to the following definition:

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Definition 5 (1PI). A connected graph is 1-particle-irreducible or 2-connected if any edge can be removed without decomposing the graph into two connected components wich are mutually disconnected.

1.5.2 Feynman graphs

Definition of a Feynman graph A Feynman graph Γ formally consists of

1. a finite set of half-edges Γ^he

2. a partition of Γ^he into a set of vertices V_Γ

3. a set of disjoint unordered pairs of half-edgesE_Γ making up the internal edges.

The external edges of Γ are the half-edges not belonging to an internal edgee∈E_Γ, Γ^ext = Γ^he\EΓ.

A Feynman graph automorphism is a bijection f : Γ^he→Γ^he, respecting verticesv, inner edges eand external edges h,

v∈V_Γ⇒f(v)∈V_Γ e∈E_Γ⇒f(e)∈E_Γ h∈Γ^ext⇒f(h) =h.

The automorphisms form a group Aut(Γ) =n

f

f is an automorphism Γ^he→Γ^heo .

Definition 6. The symmetry factor of a Feynman graph Γ is the number of automorphisms it permits,

sym(Γ) =|Aut(Γ)|

Feynman rules

So far, n-point functions have been correlation functions of the quantum field at different points in spacetime. In scattering experiments on the other hand one observes the momenta of the particles rather than their position. To make the connection, the n-point functions are Fourier-transformed and computed in momentum space. Hence they are functions ofndifferent momenta instead ofndifferent spacetime points. Since Dyson’s serieseq. (1.30)introduces “intermediate” spacetime points (i.e. points which are none of the xj the originaln-point function is computed for), after Fourier transformation there are undetermined momenta in the Wick-expansion eq. (1.31). These momenta correspond to closed loops in the Feynman graph. By the general principles

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of quantum mechanics, one needs to integrate over any unobserved quantity. Therefore a Feynman graph with loops contains one (4-dimensional) integral for each independent loop it has.

The Feynman rules F are a mapping between Feynman graphs of the perturbation serieseq. (1.31)and Feynman amplitudes as functions of the external momenta. If the graph Γ is a tree, F(Γ) is the product of the individual Feynman amplitudes of its constituents, i.e. vertex- and propagator amplitudes. If Γ contains loops, F(Γ) is an integral over all undetermined internal momenta and has the structure

F(Γ) = 1 sym(Γ)

Z

d^4·|Γ|k N(Γ) Q

e∈E_Γxe

= Z

d^4·|Γ|kI(Γ). (1.32) Here, I(Γ) was introduced as a shorthand for the Feynman integrand of Γ without the integration itself. I(Γ) is the product of the individual Feynman amplitudes of the constituents of Γ and depends on both external and integration momenta (and masses and coupling parameters). The number of independent loops (i.e. the first Betti number) of Γ is |Γ| and the numerator N(Γ) depends on the underlying theory.

For scalarφ^s-theory it involves a factor (−iλ) for each vertex in V_Γ and a factor ifor each internal edge inEΓ. This prescription follows from the Lagrangian of the theory.

Especially, if there is a monomial

−λ n!·φⁿ

in the Lagrangian, there will be a corresponding monomial +_n!^λ ·φⁿ in the interaction Hamiltonian. Via the Dyson series eq. (1.30)such a monomial gives rise to a n-valent vertex with Feynman rule−iλ. Therefore, the factor c(Γ) in eq. (1.31) contains both the symmetry factor _sym(Γ)¹ and the numerator N(Γ).

Whenever an expression like Γ = f(p, . . .) appears, we implicitly mean F(Γ) = f(p, . . .), the Feynman rules F are the only mapping between graphs and functions appearing in this work. The integrandI(Γ) however will be denoted explicitly.

Euler characteristic

The Euler characteristic [Wei18b] relates the number of edges, loops and vertices of a plane graph. If Γ is a plane connected graph and|E_Γ|the number of its (internal) edges and|V_Γ|the number of its vertices and then

|V_Γ| − |E_Γ|+|Γ|= 1 (1.33) Especially, a tree does not contain any loops, hence a plane connected treeT has one more vertex than it has edges,

|V_T| − |E_T|= 1. (1.34) 1.5.3 Renormalization

Feynman integrals as obtained from the Feynman rules eq. (1.32) are generally divergent. Technically the divergences arise from the ill-definedness of the replacement

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eq. (1.21) to obtain pointlike quantum fields but can also be understood physically:

In an interacting theory, there is self-interaction of the fields at any time. Doing a computation in the standard perturbative way as outlined insection 1.5.1 ignores this phenomenon by assuming the fields propagate as free fields between pointlike interaction events. Therefore the divergences in unrenormalized quantum field theory are an indicator that the quantities computed there are meaningless because they ground on a misconception of interaction. Renormalization can be understood to correct this, see [Kla16, chapter 16.3]. A theory is called renormalizable if the superficial degree of divergence of a Feynman graph only depends on the number (and, if different types of fields are present, on the types) of external edges. For a renormalizable theory, the renormalization procedure eliminates all infinities in a consistent methodical manner.

Motivated by classical (i.e. non-quantum) field theories, it seems plausible to write a Lagrangian density for an interacting φ^s theory as

L= 1

2∂_µφ∂^µφ−1 2m²φ²

| {z }

L0

−λ s!φ^s

| {z }

Lint

. (1.35)

But to fully include the self-interactions of a quantized field, it turns out one has to use renormalized quantities

φ_r= 1

√Zφ λ_r= Z(λ_r)^s²

Z_λ(λ_r)λ m_r =

pZ(λr)

pZm(λr)m. (1.36) The Z-factors are themselves functions of λr. The free field (which does not need renormalization) is restored in the limit λ → 0 by the conditions Z(0) = 1, Z_λ(0) = 1, Z_m(0) = 1. Inserting eq. (1.36)turns eq. (1.35)into

L= 1

2Z∂µφr∂^µφr−1

2Zmm²_rφ²_r−Zλλr

s! φ^s_r (1.37)

= 1

2∂_µφ_r∂^µφ_r−1 2m²_rφ²_r

| {z }

Lr,0

−λ_r s!φ^s_r+1

2(Z−1)∂_µφ_r∂^µφ_r−1

2(Z_m−1)m²_rφ²_r−(Z_λ−1)λ_r s! φ^s_r

| {z }

Lr,int

.

The part L_r,0 equals the unrenormalized free Lagrangian L₀ from eq. (1.35)with the only difference that φ_r and m_r take the role of φ and m. On the contrary, L_r,int contains three additional free parameters Z(λ_r), Z_m(λ_r), Z_λ(λ_r) which formally give rise to further types of interactions, i.e. additional vertices in Feynman graphs.

The renormalized interaction Hamiltonian density in the interaction picture (i.e. as a function of the interaction picture fieldsφ_r,0(t,x) as ineq. (1.27)) reads

H_r,int(t,x) =−1

2(Z−1)∂µφ_r∂^µφ_r+1

2(Zm−1)m²_rφ²_r+Z_λλ_r s! φ^s_r.

Consequently, Dyson’s serieseq. (1.30)also includesZ-factors. Since theZ-factors are functions ofλ_r,

Z = 1 +

∞

X

n=1

z_nλⁿ_r Z_λ= 1 +

∞

X

n=1

z_λ,nλⁿ_r Z_m = 1 +

∞

X

n=1

z_m,nλⁿ_r,

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eventually Dyson’s series again has one single expansion paramter, namely λr. This procedure turns out to be feasible in many cases, especially in the standard model of particle physics.

For a renormalizable theory it is possible to choose the coefficientszn, zλ,n and zm,n

in every order λⁿ_r such that the Feynman integrals become finite. There are different possible renormalization schemes (eventually yielding the same physical result) [Col03].

We will use kinematic renormalization. Let F(Γ)(p) =

Z

d⁴kI(Γ)(p, k)

be a logarithmic divergent Feynman integral without internal subdivergences, wherep is an external momentum. Kinematic renormalization amounts to choosing some fixed valuep₀ and computing the renormalized Feynman integral

FR(Γ)(p, p₀) = Z

d⁴k

I(Γ)(p, k)− I(Γ)(p₀, k)

= Z

d⁴kIR(Γ)(p, p₀, k). (1.38) A graph with worse than logarithmic divergence behaviour requires the subtraction of derivatives ofI(Γ), that is, the physical value of this quantity is fixed only up to some polynomial in the external momenta. In renormalizable theories, the degree of this polynomial matches the degree (in derivatives) of monomials in the Lagrangian density, allowing the ambiguity to be eliminated by finite redefinition of the renormalization factorsZi. If not more than two derivatives appear inL(as is the case for the Lagrangian eq. (1.37)we will be using), graphs may be at most quadratically divergent for the theory to be renormalizable.

If the integral involves divergent sub-integrals, a single subtraction is not sufficient to render it finite. Instead, these subintegrals need to be renormalized first and then their renormalized version is used in the outer integral. But even then a renormalized integrand I_R(Γ) can be written down explicitly, it consists of sums and products of differences of the individual subintegrands. The combinatorics of this procedure is fully understood and encoded in a Hopf algebra [Kre98;KW99].

Kinematic renormalization has the salient property that any Feynman integral which does not depend on external momenta is renormalized to zero. This can be understood from the simplest example eq. (1.38): In case the integrand I(Γ) does not depend on any external momentum, the very same integrand is subtracted and the renormalized integrand is zero. For a more complicated Feynman integral, it is sufficient if there is a single subintegral which does not involve external momenta for the whole integral to vanish. Such a subintegral arises as soon as there is a closed path of edges which is connected with the rest of the graph in just a single vertex. This shape is dubbed tadpole after the similar looking animal.

It is possible and consistent with the Hopf algebra of renormalization to leave out any tadpole graph from the beginning. This way using kinematic renormalization signifi- cantly reduces the number of 1PI Feynman graphs to be considered for a given external leg structure and loop number.

For later use, the following 1-loop integrals evaluate to zero in kinematic renormal-

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ization because they belong to tadpole graphs : Z d⁴k

(2π)²1−→^ren. 0 Z d⁴k

(2π)² 1 xk

−→ren. 0 (1.39)

Z d⁴k (2π)²

1 x_k+p

−→ren. 0 for any fixed p.

For two loops, if u, v and w are arbitrary linear combinations of integration variables and external momenta,

Z Z d⁴kd⁴l (2π)⁴

xu

xvxw

−→ren. 0. (1.40)

Essentially, the vanishing is due to the fact that if there are only two offshell variables x in the denominator, a linear transformation can always render them independent of external momenta. This is representing the graphical fact that every 2-loop graph which has only two internal edges is a tadpole due to the Euler characteristicsection 1.5.2.

1.6 Diffeomorphisms

In general, a diffeomorphism is a bijection f between manifolds such that f and f⁻¹ are differentiable. In our context of field theory, a diffeomorphism is a functionf (not dependent on x) which relates one fieldρ(x) to another field

φ(x) =f(ρ(x))

at the same spacetime point x. For this locality, diffeomorphisms are also called point transformations.

In classical mechanics, point transformations are a subset of canonical transformations of the generalized positions and momenta. In quantum theory, it seems point transformations are given by unitary operators but this operator has “a form of infinite series and, in general, will be never convergent and so never unitary as it is”[Nak55, p.384]. Therefore it is not at all clear that diffeomorphisms of quantum fields are a unitary transformation of these fields.

1.7 Motivation for this work

Perturbative scattering theory in the interaction picture in its most basic form as outlined insection 1.5.1 relies on the assumption that for infinite past or future the states of an interacting quantum field theory resemble those of a free theory. In terms of scattering states|ii,|fifromeq. (1.28)this translates to the assumption that “sufficiently”

far from the interaction point, particles propagate as free particles.

This concept is usually introduced qualitatively in the form of physical intuition.

One possible way is by using an interaction Hamiltonian which is switched on and off in

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time, and goes under the nameadiabatic hypothesis [Das10, p. 229]. But in the light of Haag’s theorem (1.2) it is questionable how an interacting and a non-interacting particle can possibly propagate in the same way. If their propagators, i.e. 2-point functions, coincide, both fields are necessarily free. Also the compatibility of Z-factors with the adiabatic hypothesis rises questions [BD67, sec. 16.4].

A second, probably even more severe problem is a core ingredient in the construction of the interaction picture, namely the assumption that for some time (in our case eq. (1.25) t = 0 has been chosen) the free fields of the interaction picture φ₀ coincide with the interacting Heisenberg fields φ. Since Haag’s theorem dictates there is no unitary transformation between free and interacing fields, the operatoreq. (1.26)cannot be unitary [Kla16].

Althought similar, these two problems are not the same: It is possible to solve the first problem by carefully establishing a notion of asymptotic states which for no time are unitarily related to free fields (Haag-Ruelle theory, an overview is given in [Fra06]).

But this does not cure the second problem because the interaction picture relies on the relation eq. (1.25) no matter what the asymptotic states are. Existence of the interaction picture however is crucial for perturbative quantum field theory. Even if asymptotic states are used which do not violate Haag’s theorem, it is unclear how to obtain actual predictions for them if the machinery of Feynman graphs is not available.

On the other hand, perturbative quantum field theory is highly successfull at predicting and reproducing experimental data. As one single example, consider the correction ae = ¹₂(g−2) of the magnetic moment g of the electron [HFG08]. The best currently available experimental value [TP18] is a_e,measured = (1159652180.91±0.26)·10⁻¹². An approximation of a_e available from perturbative quantum field theory from 2012 [Aoy+12] is ae,qft = (1159652181.13±1.27)·10⁻¹², it coincides with the experimental value within the given accuracy. The 4-loop integrals from quantum electrodynamics involved ina_e,qft have been computed numerically to great precision recently [Lap17].

A possible remedy lies in the use of quantum field diffeomorphisms: As was demon- strated in [KY17], there is (at least formally) a diffeomorphism ρ(φ) of an interacting quantum fieldφsuch that the propagator ofρis free for a given fixed offshell momentum.

At the same time, the S-matrix ofρ equals the one ofφ. In this work, we will prove the latter statement and explicitly construct a suitable diffeomorphism for the former. In a profound analysis which is not pursued in this work, it might then be possible to refor- mulate the asymptotic conditions for scattering states in terms of this diffeomorphism and eliminate (contradictory) assumptions about the interaction Hamiltonian. For its implications regarding the adiabatic hypothesis, we call this special diffeomorphism

“adiabatic”. It is so far completely unclear if the adiabatic diffeomorphism can also be utilized to approach the second problem, namely the construction of an “interaction picture” without unitary equivalence between free and interacting fields.

A second motivation to study quantum field diffeomorphisms lies in the fact that these diffeomorphisms are generally perturbatively non-renormizable but still their S- matrix coincides with the one of a renormalizable theory. Since the diffeomorphism can be tuned arbitrarily, it might be possible to understand how to extract finite S- matrix elements even from a non-renormalizable theory in special cases. Especially, the degree of divergence of Feynman graphs for field diffeomorphisms is proportional to the loop number, this phenomenon also appears in the (not yet fully understood) quantum

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theory of gravity, comparesection 2.1.3.

The third motivation for this work arises from path integrals [Fey48]. The path integral formalism is an alternative formulation of quantum field theory which we will not use here. A priori, the path integral is invariant under any diffeomorphisms of the field [ACO01], at least if corrections for non-commuting terms are taken into account [Omo77]. Therefore it is worth checking if and to what extent this invariance also exists in the canonical perturbative formulation of quantum field theory outlined in section 1.5.

Finally, has been shown on the level of the Lagrangian density that the S-matrix is invariant under point transformations in renormalized perturbation theory in coordinate space [Flu75]. We consider Feynman graphs, i.e. the explicit terms arising in the perturbation series. Thereby we verify that the formal manipulations of the Lagrangian density are consistent with the perturbative expansion.

1.8 Organization of the text

This text is organized as follows:

Chapter 2 reviews what is already known on diffeomorphisms of free quantum fields and fixes notation. The central result there is that diffeomorphisms of free fields do not alter the S-matrix, this implies they are not observable in scattering experiments.

Chapter 3represents the core part of this text, it discusses in great detail diffeomorphisms of scalar φ³-theory. We establish that applying a diffeomorphism to the field does not change the S-matrix even if the underlying field has interaction. This allows to regard all quantum field theories which are related by diffeomorphisms as physically equivalent. Hence it is justified to pick a special field out of this class which has desir- able properties. We construct explicitly to all orders perturbation theory an “adiabatic diffeomorphism” which renders the time ordered offshell 2-point function free for a fixed momentum.

Chapter 4essentially repeats the steps fromchapter 3for a scalar theory withφ^s-type interaction. Again, diffeomorphisms do not alter the S-matrix and it turns out that the adiabatic diffeomorphism can be given explicitly even in this case.

Appendix A is used to store a variety of lemmas which we need throughout the text. They are not given (and proved) in the main text because this would diminish readability.

Appendix B is a summary of the whole thesis in colloquial german language. It is intended for the reader who has no background in physics and does not contain any additions relevant for the main text.

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2 Free theory

Chapter 2follows sections 3 and 4 of [KY17]. Note that the fieldρused in the following is what is called φ in [KY17]. An earlier investigation of free field diffeomorphisms is [KV13].

2.1 Diffeomorphism of a free scalar theory

2.1.1 Lagrangian density

A free scalar quantum field theory is defined by its (unrenormalized) Lagrangian density L(x) = 1

2∂µφ(x)∂^µφ(x)−1

2m²φ²(x). (2.1)

The first summand can be written in a different form using partial integration. This produces an additional surface term, but imposing either periodic boundary conditions or fields decaying at infinity eliminates this contribution and we arrive at

L(x) =−1

2φ(x)∂_µ∂^µφ(x)−1

2m²φ²(x). (2.2)

The quantum fieldρ is defined implicitly via the diffeomorphism φ(x) =ρ(x) +a1ρ²(x) +a2ρ³(x) +. . .=

∞

X

j=0

ajρ^j+1(x), aj ∈R. (2.3) We defined a0 = 1 and excluded a constant shift, this means the diffomorphism is tangent to the identity map.

Inserting eq. (2.3)intoeq. (2.2)yields the Lagrangian density for the fieldρ:

L=−1 2





∞

X

j=0

ajρ^j+1



∂µ∂^µ

∞

X

k=0

akρ^k+1

!

−1 2m²

∞

X

l=0

alρ^l+1

∞

X

m=0

amρ^m+1

=−1 2

∞

X

j=0

∞

X

k=0

a_ja_kρ^j+1∂_µ∂^µρ^k+1−1 2m²

∞

X

l=0

∞

X

m=0

a_la_mρ^l+1ρ^m+1. (2.4) Under the spacetime integral to compute the action from the Lagrangian density, any total divergence vanishes and we can identify such terms with zero. The total divergence

0 =∂µ(∂^µρ·ρⁿ) =∂µ∂^µρ·ρⁿ+n∂^µρ·ρⁿ⁻¹∂µρ allows to set

∂^µρ∂µρ·ρⁿ=− 1

n+ 1ρⁿ⁺¹·∂µ∂^µρ (2.5)

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Usingeq. (2.5), the summand in the first term of eq. (2.4)becomes ρ^j+1∂µ∂^µρ^k+1 = (k+ 1)

−k 1

j+k+ 1ρ^j+k+1∂µ∂^µρ+ρ^j+k+1∂µ∂^µρ

= (j+ 1)(k+ 1)

j+k+ 1 ρ^j+k+1∂µ∂^µρ.

We rearrange summations in eq. (2.4) by introducing n =j+k+ 1 instead of j and n=m+l+ 2 instead of m:

L=−1 2

∞

X

n=1 n−1

X

k=0

an−k−1a_k(n−k)(k+ 1)

n ρⁿ∂_µ∂^µρ−1 2m²

∞

X

n=2 n−2

X

k=0

an−k−2a_kρⁿ. (2.6) The nested sums in the Lagrangian density of ρcan be written in a more systematic form by using Fa`a di Bruno’s formula eq. (1.19). The first derivative of φis

∂_µφ=

∞

X

n=0

a_n∂_µρⁿ⁺¹ =

∞

X

n=0

(n+ 1)anρⁿ·∂_µρ=φ⁰·∂_µρ.

Consequently, the kinetic term ineq. (2.1)amounts to 1

2∂_µφ∂^µφ= 1

2∂_µρ∂^µρ· φ⁰2

. (2.7)

We write the outer derivativeφ⁰ in a form where the constant contribution is extracted, φ⁰=

∞

X

n=0

(n+ 1)!an

ρⁿ

n! =a0ρ⁰+

∞

X

n=1

(n+ 1)!an

ρⁿ

n! = 1 +ϕ⁰.

Now Fa`a di Bruno’s formulaeq. (1.19)can be applied to compute (ϕ⁰)². We setg(ρ) :=

ϕ⁰(ρ) and f(t) :=t², i.e. g_n= (n+ 1)!a_n and f_n=n!δ_n2, it follows f(g(ρ)) = ϕ⁰(ρ)2

=

∞

X

n=1

2·Bn,2(2!a1,3!a2,4!a3, . . .)ρⁿ n!, φ⁰2

= 1 + 2ϕ⁰+ ϕ⁰2

= 1 +

∞

X

n=1

2(n+ 1)!an+ 2Bn,2(2!a1,3!a2. . .) n!

ρⁿ.

The first summand in the parenthesis can also be written in terms of Bell Polynomials (n+ 1)!an=Bn,1(2!a1,3!a2, . . .) =Bn+1,1(1!a0,2!a1,3!a2, . . .),

φ⁰2

= 1 +

∞

X

n=1

2Bn+1,1(1!a0,2!a1,3!a2, . . .) +Bn,2(2!a1,3!a2, . . .)

n! ρⁿ. (2.8)

Ifeq. (2.5)is applied to eq. (2.8), the kinetic term eq. (2.8)takes the form 1

2∂_µφ∂^µφ= 1

2∂_µρ∂^µρ+

∞

X

n=1

Bn,1(2!a1,3!a2, . . .) +Bn,2(2!a1,3!a2, . . .)

n! ∂_µρ∂^µρ·ρⁿ

=−1

2ρ∂_µ∂^µρ−

∞

X

n=3

Bn−2,1(2!a₁,3!a₂, . . .) +Bn−2,2(2!a₁,3!a₂, . . .)

(n−1)! ρⁿ⁻¹∂_µ∂^µρ.

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The mass term ofeq. (2.1)can also be treated with Fa`a di Bruno’s formulaeq. (1.19).

We set g(ρ) =φ(ρ) andf(t) =t², then f(g(ρ)) =φ(ρ)² =

∞

X

n=2

2·B_n,2(1!a₀,2!a₁,3!a₂, . . .)ρⁿ n!. The Lagrangian density ofρ is

L=−1

2ρ∂µ∂^µρ−

∞

X

n=3

Bn−2,1(2!a1,3!a2, . . .) +Bn−2,2(2!a1,3!a2, . . .)

(n−1)! ρⁿ⁻¹∂µ∂^µρ

−

∞

X

n=2

B_n,2(1!a₀,2!a₁,3!a₂, . . .)

n! ρⁿ. (2.9)

2.1.2 Feynman rules

To obtain the vertex Feynman rules from eq. (2.9)we consider the two sums one after another. First, for any n≥3, there is a term

Bn−2,1(2!a1,3!a2, . . .) +Bn−2,2(2!a1,3!a2, . . .)

(n−1)! ρⁿ⁻¹∂µ∂^µρ (2.10)

In momentum space each derivative ∂_µ becomes a momentum ip_µ. Also, we sum over all permutations of the external legs, this gives a sum over the external momenta and for each summand a factor (n−1)! for the possible permutations of the remaining (n−1) legs. Finally, there is a factorifrom the perturbation expansioneq. (1.30). The Feynman rule foreq. (2.10) is an-valent Vertex with amplitude

i

Bn−2,1(2!a1,3!a2, . . .) +Bn−2,2(2!a1,3!a2, . . .)

p²₁+p²₂+. . .+p²_n

=idn−2

2

p²₁+p²₂+. . .+p²_n

.

We have introduced a parameter dn consistent with [KY17],

d_n= 2(B_n,1(2!a₁,3!a₂, . . .) +B_n,2(2!a₁,3!a₂, . . .)), n≥1, d₀ = 1. (2.11) If the explicit formula fromdefinition 3or the form eq. (2.6)is used, one obtains

d_n=n!

n

X

k=0

a_n−ka_k(n−k+ 1)(k+ 1). (2.12) The second term in eq. (2.9) represents a n-valent vertex. Multiplying with the symmetry factorn! produces the Feynman amplitude

−iB_n,2(1!a0,2!a1,3!a2, . . .) =im²cn−2. The factorcn is in accordance with [KY17],

cn=−B_n+2,2(1!a0,2!a1,3!a2, . . .) =−1

2(n+ 2)!

n

X

l=0

a_lan−l. (2.13)

The propagator and diffeomorphisms of an interacting field theory