Mass and strong interaction.

DW

Hello,
Several months ago, Frank Wilczek wrote two
articles about F=ma (appearing in "Physics Today"
Oct and Dec, 2004). He thinks it's a culture to
accept such a formula. It seems to me he tried to
understand the origin inertia mass from the view
point of strong interaction. I am just curious whether we
can trace the origin of mass back to the interaction
of elementary particles. In other words, why we need some
force to accelerated something (e.g a neutron) in vacuum from
the view point of QFT? Is it due to some kind of interaction
preventing the acceleration of the neutron? Or is it because
of the structure of space-time? Is this related to Higgs?

Thanks!
DW

Igor Khavkine

On 2005-06-23, DW <an_eyas@yahoo.com.cn> wrote:
> Hello,
> Several months ago, Frank Wilczek wrote two
> articles about F=ma (appearing in "Physics Today"
> Oct and Dec, 2004). He thinks it's a culture to
> accept such a formula. It seems to me he tried to
> understand the origin inertia mass from the view
> point of strong interaction. I am just curious whether we
> can trace the origin of mass back to the interaction
> of elementary particles. In other words, why we need some
> force to accelerated something (e.g a neutron) in vacuum from
> the view point of QFT? Is it due to some kind of interaction
> preventing the acceleration of the neutron? Or is it because
> of the structure of space-time? Is this related to Higgs?

>From the point of view of QFT, inertial mass is the minimum energy m
between the ground state and an excited state of a quantum field.
Sometimes m is also referred to as a gap. A single particle corresponds
to a small excitation on top of the ground state. Think first excited
state of a harmonic oscillator compared to its ground state.

Relativistic invariance these states be labeled by momentum and the
energy be E=sqrt(p^2+m^2) or E^2-p^2 = m^2. This translates into single
particle states being described by solutions of a wave equation of the
form box(phi) = 0, where "box" is the d'Alambertian. The type of wave
function phi depends on the type of quantum field under consideration.
Locality (or rather cluster separability) garantees that several
particle states containing widely separating wave packets obey the same
equation locally.

Once interactions are taken into account, the joint wave equation
describing a multi-particle system can no longer be factored into
individual wave equations and becomes coupled (think multiparticle
interacting Schroedinger equation).

The classical limit is obtained by applying the WKB approximation, which
tells us that the phase of the wave function satisfies the
Hamilton-Jacobi equation. The characteristics of this first order scalar
PDE are the trajectories of the solution of the relativistic Hamiltonian
dynamical system with Hamiltonian of the form

H = (E^2-p^2 - m^2) + (other particle) + (interactions) = 0.

If the non-relativistic limit is taken, the Hamiltonian formulation
turns into the more familiar

H = p^2/2m + (other particles) + (interactions),

where p is now the non-relativistic momentum p = mv. From here Newton's
second law F(due to interactions) = dp/dt = m dv/dt is recovered in the
usual way.

To review. The m in the F=ma comes from the relativistic relation
E^2-p^2 = m^2. This relation comes from the wave equation satisfied by
few particle states. This wave equation is derived from the spectrum of
the excitations of a quantum field. The specific form of the spectrum is
due to relativistic invariance and the existence of a minimum excitation
energy (the gap).

The next question is "why the gap"? And the answer is not really known.
It is hypothesized that all elementary particles in the standard model
start out as massless (no gap), but then acquire a mass of the form
"(expectation value of Higgs field) x (coupling constant to Higgs field)".

Hope this helps.

Igor

Eugene Stefanovich

DW wrote:
> In other words, why we need some
> force to accelerated something (e.g a neutron) in vacuum from
> the view point of QFT? Is it due to some kind of interaction
> preventing the acceleration of the neutron? Or is it because
> of the structure of space-time? Is this related to Higgs?

This is called "Newton's 1-st law": a particle does not
accelerate (moves with a constant velocity) if there are no
other particles around which can interact with it.

If there are other particles around, then normally the first
particle feels a force, i.e., accelerates. Particle's mass
m connects acceleration with force. Most naturally, parameter
m (along with spin) appears in the Wigner's classification
scheme of elementary particles as unitary irreducible
representations of the Poincare group.

Eugene Stefanovich.

Eugene Stefanovich

Igor Khavkine wrote:
> On 2005-06-23, DW <an_eyas@yahoo.com.cn> wrote:
>
>>Hello,
>> Several months ago, Frank Wilczek wrote two
>>articles about F=ma (appearing in "Physics Today"
>>Oct and Dec, 2004). He thinks it's a culture to
>>accept such a formula. It seems to me he tried to
>>understand the origin inertia mass from the view
>>point of strong interaction. I am just curious whether we
>>can trace the origin of mass back to the interaction
>>of elementary particles. In other words, why we need some
>>force to accelerated something (e.g a neutron) in vacuum from
>>the view point of QFT? Is it due to some kind of interaction
>>preventing the acceleration of the neutron? Or is it because
>>of the structure of space-time? Is this related to Higgs?
>
>
>>From the point of view of QFT, inertial mass is the minimum energy m
> between the ground state and an excited state of a quantum field.
> Sometimes m is also referred to as a gap. A single particle corresponds
> to a small excitation on top of the ground state. Think first excited
> state of a harmonic oscillator compared to its ground state.
>
> Relativistic invariance these states be labeled by momentum and the
> energy be E=sqrt(p^2+m^2) or E^2-p^2 = m^2. This translates into single
> particle states being described by solutions of a wave equation of the
> form box(phi) = 0, where "box" is the d'Alambertian. The type of wave
> function phi depends on the type of quantum field under consideration.
> Locality (or rather cluster separability) garantees that several
> particle states containing widely separating wave packets obey the same
> equation locally.
>
> Once interactions are taken into account, the joint wave equation
> describing a multi-particle system can no longer be factored into
> individual wave equations and becomes coupled (think multiparticle
> interacting Schroedinger equation).
>
> The classical limit is obtained by applying the WKB approximation, which
> tells us that the phase of the wave function satisfies the
> Hamilton-Jacobi equation. The characteristics of this first order scalar
> PDE are the trajectories of the solution of the relativistic Hamiltonian
> dynamical system with Hamiltonian of the form
>
> H = (E^2-p^2 - m^2) + (other particle) + (interactions) = 0.
>
> If the non-relativistic limit is taken, the Hamiltonian formulation
> turns into the more familiar
>
> H = p^2/2m + (other particles) + (interactions),
>
> where p is now the non-relativistic momentum p = mv. From here Newton's
> second law F(due to interactions) = dp/dt = m dv/dt is recovered in the
> usual way.
>
> To review. The m in the F=ma comes from the relativistic relation
> E^2-p^2 = m^2. This relation comes from the wave equation satisfied by
> few particle states. This wave equation is derived from the spectrum of
> the excitations of a quantum field. The specific form of the spectrum is
> due to relativistic invariance and the existence of a minimum excitation
> energy (the gap).
>
> The next question is "why the gap"? And the answer is not really known.
> It is hypothesized that all elementary particles in the standard model
> start out as massless (no gap), but then acquire a mass of the form
> "(expectation value of Higgs field) x (coupling constant to Higgs field)".

Expanding on my previous post let me suggest a different logic of
introducing mass, force, and acceleration in QFT.

1. Elementary particles are described by unitary irreducible
representations of the Poincare group. They are characterized by
two parameters (according to two Casimir operators of the Poincare
group): m (mass) and s (spin). Wigner, 1939.

2. Particle position is described by the Newton-Wigner (1949) operator
r. In a multiparticle non-interacting system, the commutators
of r and p (momentum) with the Hamiltonian H_0 are

[r, H_0] = iv
[p, H_0] = 0

so that

r(t) = exp(iHt) r(0) exp(-iHt) = r + vt
p(t) = p(0)

i.e., each particle moves with constant velocity
(v = p/sqrt(p^2 + m^2)), which is Newton's 1st law.

3. In the presence of interaction, the multiparticle Hamiltonian
is H = H_0 + V (Dirac, 1949), where V depends on positions
and momenta of all
particles in the system (ignore for a moment creation/annihilation
terms, they don't change anything, in principle, see below).
The time derivative of p then becomes

dp/dt = -i [p, H] = dV/dr (1)

The operator F = dV/dr is called "force" (by definition).

4. In the non-relativistic limit p = mv, and eq. (1) yields

a = dv/dt = F/m

which is the 2nd Newton's law (a is acceleration).

5. Creation and annihilation of particles are described by
terms in V that do not commute with the particle number operator.
They can be added to eq. (1). The most prominent effect would
be the appearance of "radiation reaction" forces due to emission
of photons by accelerated charges.

Eugene.